Search: a004396 -id:a004396
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5, 10, 17, 20, 21, 23, 29, 34, 40, 42, 43, 46, 53, 58, 65, 68, 69, 71, 77, 80, 81, 83, 84, 85, 86, 87, 89, 92, 93, 95, 101, 106, 113, 116, 117, 119, 125, 130, 136, 138, 139, 142, 149, 154, 160, 162, 163, 166, 168, 169, 170, 171, 172, 174, 175, 178, 184, 186, 187
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OFFSET
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0,1
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LINKS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A002264
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Nonnegative integers repeated 3 times.
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+10
122
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0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25
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OFFSET
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0,7
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COMMENTS
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Chvátal proved that, given an arbitrary n-gon, there exist a(n) points such that all points in the interior are visible from at least one of those points; further, for all n >= 3, there exists an n-gon which cannot be covered in this fashion with fewer than a(n) points. This is known as the "art gallery problem". - Charles R Greathouse IV, Aug 29 2012
The inverse binomial transform is 0, 0, 0, 1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729,.. (see A000748). - R. J. Mathar, Feb 25 2023
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LINKS
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FORMULA
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a(n) = floor(n/3).
a(n) = (3*n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3)))/9. - Hieronymus Fischer, Sep 18 2007
Complex representation: a(n) = (n - (1 - r^n)*(1 + r^n/(1 - r)))/3 where r = exp(2*Pi/3*i) = (-1 + sqrt(3)*i)/2 and i = sqrt(-1). - Hieronymus Fischer, Sep 18 2007; - corrected by Guenther Schrack, Sep 26 2019
For n >= 3, a(n) = floor(log_3(3^a(n-1) + 3^a(n-2) + 3^a(n-3))). - Vladimir Shevelev, Jun 22 2010
a(n) = n - 2 - a(n-1) - a(n-2) for n > 1 with a(0) = a(1) = 0. - Derek Orr, Apr 28 2015
a(n) = a(n-1) + a(n-3) - a(n-4), n > 4.
a(n) = (n - 1 + 0^((-1)^(n/3) - (-1)^n) - 0^((-1)^(n/3)*(-1)^(1/3) + (-1)^n))/3. (End)
a(n) = (3*n - 3 + r^n*(1 - r) + r^(2*n)*(r + 2))/9 where r = (-1 + sqrt(-3))/2. - Guenther Schrack, Sep 26 2019
E.g.f.: exp(x)*(x - 1)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022
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MAPLE
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MATHEMATICA
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Flatten[Table[{n, n, n}, {n, 0, 25}]] (* Harvey P. Dale, Jun 09 2013 *)
Table[Floor[n/3], {n, 0, 20}] (* ~~~ *)
Table[(n - Cos[2 (n - 2) Pi/3] + Sin[2 (n - 2) Pi/3]/Sqrt[3] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n - 2, -1/2] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 0, 0, 1}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[x^3/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)
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PROG
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(Haskell)
a002264 n = a002264_list !! n
a002264_list = 0 : 0 : 0 : map (+ 1) a002264_list
(PARI) v=[0, 0]; for(n=2, 50, v=concat(v, n-2-v[#v]-v[#v-1])); v \\ Derek Orr, Apr 28 2015
(Magma) &cat [[n, n, n]: n in [0..30]]; // Bruno Berselli, Apr 29 2015
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CROSSREFS
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Cf. A001477, A002265, A002266, A004526, A008615, A008620, A010761, A010762, A010872, A010873, A010874, A022003, A110532, A110533, A137221 (binom. Trans).
Apart from the zeros, this is column 3 of A235791.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A032766
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Numbers that are congruent to 0 or 1 (mod 3).
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+10
112
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0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103
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OFFSET
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0,3
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COMMENTS
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Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - Hans Havermann, May 26 2002
Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - Gary Detlefs, Mar 19 2010
Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - Robert Price, May 30 2013
a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013
Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015
Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - Bruno Berselli, Dec 09 2015
For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015
Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016
Also the clique covering number of the n-Andrásfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018
Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022
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LINKS
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Eric Weisstein's World of Mathematics, Web Graph
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FORMULA
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G.f.: x*(1+2*x)/((1-x)*(1-x^2)).
a(n) = (6n - 1 + (-1)^n)/4.
a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.
a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.
a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)
a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021
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MAPLE
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a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # Zerinvary Lajos, Mar 16 2008
select(n->member(n mod 3, {0, 1}), [$0..103]); # Peter Luschny, Apr 06 2014
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MATHEMATICA
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a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, Mar 28 2011 *)
Flatten[{#, #+1}&/@(3Range[0, 40])] (* or *) LinearRecurrence[{1, 1, -1}, {0, 1, 3}, 100] (* or *) With[{nn=110}, Complement[Range[0, nn], Range[2, nn, 3]]] (* Harvey P. Dale, Mar 10 2013 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
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PROG
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(PARI) {a(n) = n + n\2}
(Haskell)
(PARI) concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015
(SageMath) [int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024
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CROSSREFS
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Column 1 (the second leftmost) of triangular table A026374.
Column 1 (the leftmost) of square array A191450.
Cf. A253888 and A254049 (permutations of this sequence without the initial zero).
Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A004523
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Two even followed by one odd; or floor(2n/3).
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+10
65
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0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46
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OFFSET
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0,4
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COMMENTS
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Guenther Rosenbaum showed that the sequence represents the optimal number of guesses in the static Mastermind game with two pegs. Namely, the optimal number of static guesses equals 2k, if the number of colors is either (3k - 1) or 3k and is (2k + 1), if the number of colors is (3k + 1), k >= 1. - Alex Bogomolny, Mar 06 2002
a(n+1) is the maximum number of wins by a team in a sequence of n basketball games if the team's longest winning streak is 2 games. See example below. In general, floor(k(n+1)/(k+1)) gives the maximum number of wins in n games when the longest winning streak is of length k. - Dennis P. Walsh, Apr 18 2012
Sum_{n>=2} 1/a(n)^k = Sum_{j>=1} Sum_{i=1..2} 1/(i*j)^k = Zeta(k)^2 - Zeta(k)*Zeta(k,3), where Zeta(,) is the generalized Riemann zeta function, for the case k=2 this sum is 5*Pi^2/24. - Enrique Pérez Herrero, Jun 25 2012
a(n) is the pattern of (0+2k, 0+2k, 1+2k), k>=0. a(n) is also the number of odd integers divisible by 3 in ]2(n-1)^2, 2n^2[. - Ralf Steiner, Jun 25 2017
a(n) is also the total domination number of the n-triangular (Johnson) graph for n > 2. - Eric W. Weisstein, Apr 09 2018
a(n) is the maximum total domination number of connected graphs with order n>2. The extremal graphs are "brushes", as defined in the links below. - Allan Bickle, Dec 24 2021
a(n) is the minimal number of ascending or descending staircase walks necessary to cover a chessboard of size n-1, for n > 1. See Ackerman and Pinchasi. - Sela Fried, Jan 16 2023
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LINKS
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FORMULA
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G.f.: (x^2 + 2*x^3 + 2*x^4 + x^5)/(1 - x^3)^2, not reduced. - Len Smiley
a(n) = floor(2*n/3).
a(0) = a(1) = 0; for n > 1, a(n) = n - 1 - floor(a(n-1)/2). - Benoit Cloitre, Nov 26 2002
a(n) = a(n-1) + (1/2)*((-1)^floor((2*n+2)/3)+1), with a(0)=0. - Mario Catalani (mario.catalani(AT)unito.it), Oct 20 2003
a(n) = Sum_{k=0..n-1} (Fibonacci(k) mod 2). - Paul Barry, May 31 2005
O.g.f.: x^2*(1 + x)/((1 - x)^2*(1 + x + x^2)). - R. J. Mathar, Mar 19 2008
a(n) = ceiling(2*(n-1)/3) = n - 1 - floor((n-1)/3). - Bruno Berselli, Jan 18 2017
a(n) = (6*n - 3 + 2*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
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EXAMPLE
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For n=11, we have a(11)=7 since there are at most 7 wins by a team in a sequence of 10 games in which its longest winning streak is 2 games. One such win-loss sequence with 7 wins is wwlwwlwwlw. - Dennis P. Walsh, Apr 18 2012
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MAPLE
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seq(floor(2n/3), n=0..75);
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MATHEMATICA
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Table[Floor[2 n/3], {n, 0, 75}]
Table[(6 n + 3 Cos[2 n Pi/3] - Sqrt[3] Sin[2 n Pi/3] - 3)/9, {n, 0, 20}] (* Eric W. Weisstein, Apr 08 2018 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 1, 2, 2}, {0, 20}] (* Eric W. Weisstein, Apr 08 2018 *)
CoefficientList[Series[x^2 (1 + x)/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 08 2018 *)
Table[If[EvenQ[n], {n, n}, n], {n, 0, 50}]//Flatten (* Harvey P. Dale, May 27 2021 *)
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PROG
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(Haskell)
a004523 n = a004523_list !! n
a004523_list = 0 : 0 : 1 : map (+ 2) a004523_list
(Magma) [Floor(2*n/3): n in [0..50]]; // G. C. Greubel, Nov 02 2017
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CROSSREFS
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Zero followed by partial sums of A011655.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A008620
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Positive integers repeated three times.
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+10
42
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1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26
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OFFSET
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0,4
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COMMENTS
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Arises from Gleason's theorem on self-dual codes: the Molien series is 1/((1-x^8)*(1-x^24)) for the weight enumerators of doubly-even binary self-dual codes; also 1/((1-x^4)*(1-x^12)) for ternary self-dual codes.
The number of partitions of n into distinct parts where each part is either a power of two or three times a power of two.
The dimension of the space of modular forms on Gamma_1(3) of weight n>0 with a(q) the generator of weight 1 and c(q)^3 / 27 the generator of weight 3 where a(), c() are cubic AGM theta functions. - Michael Somos, Apr 01 2015
a(n-1) is the minimal number of circles that can be drawn through n points in general position in the plane. - Anton Zakharov, Dec 31 2016
Number of representations n=sum_i c_i*2^i with c_i in {0,1,3,4} [Anders]. See A120562 or A309025 for other c_i sets. - R. J. Mathar, Mar 01 2023
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REFERENCES
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G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 12 Exer. 7
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Eq. (14), p. 601 and Theorem 3c, p. 602; also Problem 5 p. 620.
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LINKS
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FORMULA
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a(n) = floor(n/3) + 1.
G.f.: 1/((1-x)*(1-x^3)) = 1/((1-x)^2*(1+x+x^2)).
a(n) = Sum_{k=0..n} (k+1)*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3. (End)
The g.f. is 1/(1-V_trefoil(x)), where V_trefoil is the Jones polynomial of the trefoil knot. - Paul Barry, Oct 08 2004
E.g.f.: exp(x)*(2 + x)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022
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MAPLE
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MATHEMATICA
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Table[{n, n, n}, {n, 30}] // Flatten (* Harvey P. Dale, Jan 15 2017 *)
Table[(1 + n - Cos[2 n Pi]/3] + Sin[2 n Pi/3]/Sqrt[3])/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 1, 2}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[1/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)
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PROG
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(PARI) a(n)=n\3+1
(Haskell)
a008620 = (+ 1) . (`div` 3)
a008620_list = concatMap (replicate 3) [1..]
(Sage) def a(n) : return( dimension_modular_forms( Gamma1(3), n) ); # Michael Somos, Apr 01 2015
(Magma) a := func< n | Dimension( ModularForms( Gamma1(3), n))>; /* Michael Somos, Apr 01 2015 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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A342910
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Concatenation of all 01-words, in the order induced by A032766; see Comments.
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+10
37
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0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1
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OFFSET
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1
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COMMENTS
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Let s = (s(n)) be a strictly increasing sequence of positive integers with infinite complement, t = (t(n)).
For n >= 1, let s'(n) be the number of s(i) that are <= n-1 and let t'(n) be the number of t(i) that are <= n-1.
Define w(1) = 0, w(t(1)) = 1, and w(n) = 0w(s'(n)) if n is in s, and w(n) = 1w(t'(n)) if n is in t. Then (w(n)) is the "s-induced ordering" of all 01-words.
In the following list, W represents the sequence of words w(n) induced by A032766. The list includes five partitions and a self-inverse permutation of the positive integers.
positions in W of words w(n) such that # 0's = # 1's: A344151;
positions in W of words w(n) such that # 0's < # 1's: A344152;
positions in W of words w(n) such that # 0's > # 1's: A344153;
positions in W of words w(n) that end with 0: A344154;
positions in W of words w(n) that end with 1: A344155;
positions in W of words w(n) such that first digit = last digit: A344156;
positions in W of words w(n) such that first digit != last digit: A344157;
positions in W of words w(n) such that 1st digit = 0 and last digit 0: A344158;
positions in W of words w(n) such that 1st digit = 0 and last digit 1: A344159;
positions in W of words w(n) such that 1st digit = 1 and last digit 0: A344160;
positions in W of words w(n) such that 1st digit = 1 and last digit 1: A344161;
position in W of n-th positive integer (base 2): A344162;
positions in W of binary complement of w(n): A344163;
positions in W of palindromes: A344166;
positions in W of words such that #0's - #1's is odd: A344167;
positions in W of words such that #0's - #1's is even: A344168;
positions in W of the reversal of the n-th word in W: A344169.
For a guide to related sequences, see A341256.
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LINKS
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EXAMPLE
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The first twenty words w(n): 0, 1, 00, 01, 10, 000, 001, 11, 010, 0000, 100, 0001, 011, 101, 0010, 00000, 110, 0100, 00001, 1000.
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MATHEMATICA
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z = 250;
"The sequence s:" (* A001651, (3n/2) *)
s = Table[Floor[3 n/2], {n, 1, z}]
"The sequence t:" (* A016789; congr to 0 or 1 mod 3; *)
t = Complement[Range[Max[s]], s]
s1[n_] := Length[Intersection[Range[n - 1], s]];
t1[n_] := n - 1 - s1[n];
"The sequence s1:"
Table[s1[n], {n, 1, z}] (* A004396 *)
"The sequence t1:"
Table[t1[n], {n, 1, z}] (* A002264 *)
w[1] = {0}; w[t[[1]]] = {1};
w[n_] := If[MemberQ[s, n], Join[{0}, w[s1[n]]], Join[{1}, w[t1[n]]]]
"List tt of all binary words:"
tt = Table[w[n], {n, 1, z}] (* all the binary words *)
"All the words, concatenated:"
Flatten[tt] (* words, concatenated, A344150 *)
"Positions of words in which #0's = #1's:" (* A344151 *)
Select[Range[Length[tt]], Count[tt[[#]], 0] == Count[tt[[#]], 1] &]
"Positions of words in which #0's < #1's:" (* A344152 *)
Select[Range[Length[tt]], Count[tt[[#]], 0] < Count[tt[[#]], 1] &]
"Positions of words in which #0's > #1's:" (* A344153 *)
Select[Range[Length[tt]], Count[tt[[#]], 0] > Count[tt[[#]], 1] &]
"Positions of words ending with 0:" (* A344154 *)
Select[Range[Length[tt]], Last[tt[[#]]] == 0 &]
"Positions of words ending with 1:" (* A344155 *)
Select[Range[Length[tt]], Last[tt[[#]]] == 1 &]
"Positions of words starting and ending with same digit:" (* A344156 *)
Select[Range[Length[tt]], First[tt[[#]]] == Last[tt[[#]]] &]
"Positions of words starting and ending with opposite digits:" (* A344157 *)
Select[Range[Length[tt]], First[tt[[#]]] != Last[tt[[#]]] &]
"Positions of words starting with 0 and ending with 0:" (* A344158 *)
Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 0 and ending with 1:" (* A344159 *)
Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &]
"Positions of words starting with 1 and ending with 0:" (* A344160 *)
Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 1 and ending with 1:" (* A344161 *)
Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &]
"Position of n-th positive integer (base 2) in tt: A344162 "
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]]
"Position of binary complement of w(n): A344163"
comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]]
Table[Total[w[n]], {n, 1, 100}]
Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]]
"Palindromes:"
Select[tt, # == Reverse[#] &]
"Positions of palindromes: A344166"
Select[Range[Length[tt]], tt[[#]] == Reverse[tt[[#]]] &]
"Positions of words in which #0's - #1's is odd: A344167"
Select[Range[Length[tt]], OddQ[Count[w[#], 0] - Count[w[#], 1]] &]
"Positions of words in which #0's - #1's is even: A344168"
Select[Range[Length[tt]], EvenQ[Count[w[#], 0] - Count[w[#], 1]] &]
"Position of the reversal of the n-th word: A344169"
Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]]
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A004773
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Numbers congruent to {0, 1, 2} mod 4: a(n) = floor(4*n/3).
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+10
31
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0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90
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OFFSET
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0,3
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COMMENTS
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The sequence b(n) = floor((4/3)*(n+2)) appears as an upper bound in Fijavz and Wood.
Binary expansion does not end in 11.
Let S(n) = a(n) + a(n+1) + a(n+2). Then floor(S(n)/3) = A042968(n+1), round(S(n)/3) = a(n+1), ceiling(S(n)/3) = A042965(n+2). (End)
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LINKS
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FORMULA
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G.f.: x*(1+x+2*x^2)/((1-x)*(1-x^3)).
a(n) = (12*n-3+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
E.g.f.: (3*exp(x)*(4*x - 1) + exp(-x/2)*(3*cos((sqrt(3)*x)/2) + sqrt(3)*sin((sqrt(3)*x)/2)))/9. - Stefano Spezia, Jun 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + sqrt(2)*log(sqrt(2)+2)/4 + (2-sqrt(2))*log(2)/8. - Amiram Eldar, Dec 05 2021
a(n) = (12*n - 3 + w^(2*n)*(w + 2) - w^n*(w - 1))/9 where w = (-1 + sqrt(-3))/2.
a(n) = 2*floor(n/3) + floor((n+1)/3) + floor((n+2)/3).
a(n) = (4*n - n mod 3)/3.
a(n) = a(n-3) + 4.
a(n) = a(n-1) + a(n-3) - a(n-4).
(End)
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MAPLE
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MATHEMATICA
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a[0] = 0; a[n_] := a[n] = a[n - 1] + 2 - If[ Mod[a[n - 1], 4] < 2, 1, 0]; Array[a, 69, 0] (* Robert G. Wilson v, Dec 24 2010 *)
CoefficientList[ Series[x (1 + x + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 68}], x] (* Robert G. Wilson v, Dec 24 2010 *)
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PROG
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CROSSREFS
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Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.
Cf. A002264, A004396, A004523, A004767, A004772, A008586, A010872, A016813, A016825, A042965, A042968.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A004525
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One even followed by three odd.
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+10
25
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0, 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 19, 19, 19, 20, 21, 21, 21, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 27, 27, 28, 29, 29, 29, 30, 31, 31, 31, 32, 33, 33, 33, 34, 35, 35, 35, 36, 37, 37, 37
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OFFSET
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0,5
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COMMENTS
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a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type E_6 (binary tetrahedral group). - Paul Boddington, Oct 23 2003
(1 + x + x^2 + x^3 + x^4 + x^5) / ( (1-x^3)*(1- x^4)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(GL_2(F_3)). - N. J. A. Sloane, Jun 12 2004
The Fi1 and Fi2 sums, see A180662 for the definition of these sums, of triangle A101950 equal the terms of this sequence without the first term. - Johannes W. Meijer, Aug 06 2011
Also the domination number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017
Also the domination number of the (n-1)-Moebius laddder. - Eric W. Weisstein, Jun 30 2017
Also the rook domination number of the hexagonal hexagon board B_n [Harborth and Nienborg] - N. J. A. Sloane, Aug 31 2021
Two players play a game, the object of which is to determine a score. Player 1 prefers larger scores, while player 2 prefers smaller scores. The game begins with a set of potential scores {1,2,3, ... n}. Player 1 divides this set into two nonempty sets, one of which player 2 chooses. Player 2 the divides their chosen set into two nonempty sets, one of which player 1 chooses, and so on, until the final score is arrived at. a(n+1) is the final score when both players play optimally. - Thomas Anton, Jul 14 2023
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REFERENCES
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A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 247.
Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, New trends in algebraic geometry (Warwick, 1996), 151-233, Cambridge University Press, 1999.
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LINKS
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FORMULA
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G.f.: x*(1-x+x^2)/((1-x)^2*(1+x^2)) = x*(1-x^6)/((1-x)*(1-x^3)*(1-x^4)). - Michael Somos, Jul 19 2003
a(n) = floor(n/4) + ceiling(n/4). See also A004396, one even followed by two odd and A002620, quarter-squares: floor(n/2)*ceiling(n/2). - Jonathan Vos Post, Mar 19 2006
a(n) = Sum_{k=0..n-1} (1 + (-1)^binomial(k+1, 2))/2. - Paul Barry, Mar 31 2008
a(n) = (1/4)*(2*n - (1 - (-1)^n)*(-1)^(n*(n+1)/2)). - Bruno Berselli, Mar 13 2012
Euler transform of length 6 sequence [1, 0, 1, 1, 0, -1]. - Michael Somos, Apr 03 2017
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EXAMPLE
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G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ...
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MAPLE
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MATHEMATICA
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Table[Floor[n/4] + Ceiling[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 22 2013 *)
LinearRecurrence[{2, -2, 2, -1}, {1, 1, 1, 2}, {0, 20}] (* Eric W. Weisstein, Jun 30 2017 *)
Table[{n - 1, n, n, n}, {n, 1, 41, 2}] // Flatten (* Harvey P. Dale, Oct 18 2019 *)
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PROG
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(Maxima) makelist((1/4)*(2*n-(1-(-1)^n)*(-1)^(n*(n+1)/2)), n, 0, 75); /* Bruno Berselli, Mar 13 2012 */
(Haskell)
a004525 n = a004525_list !! n
a004525_list = 0 : 1 : 1 : zipWith3 (\x y z -> x - y + z + 1)
a004525_list (tail a004525_list) (drop 2 a004525_list)
(Python)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A357638
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Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
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+10
24
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 4, 5, 4, 1, 1, 0, 0, 0, 1, 10, 5, 4, 1, 1, 0, 0, 0, 1, 5, 13, 5, 4, 1, 1, 0, 0, 0, 0, 4, 13, 14, 5, 4, 1, 1, 0, 0, 0, 0, 1, 13, 17, 14, 5, 4, 1, 1
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OFFSET
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0,13
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COMMENTS
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We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....
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LINKS
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FORMULA
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Conjecture: The columns are palindromes with sums A298311.
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EXAMPLE
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Triangle begins:
1
0 1
0 1 1
0 1 1 1
0 0 3 1 1
0 0 1 4 1 1
0 0 1 4 4 1 1
0 0 0 4 5 4 1 1
0 0 0 1 10 5 4 1 1
0 0 0 1 5 13 5 4 1 1
0 0 0 0 4 13 14 5 4 1 1
0 0 0 0 1 13 17 14 5 4 1 1
0 0 0 0 1 5 28 18 14 5 4 1 1
Row n = 7 counts the following partitions:
. . . (322) (43) (52) (61) (7)
(331) (421) (511)
(2221) (3211) (4111)
(1111111) (22111) (31111)
(211111)
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MATHEMATICA
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skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[n], skats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]
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CROSSREFS
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Number of nonzero entries in row n appears to be A004396(n+1).
First nonzero entry of each row appears to converge to A146325.
The half-alternating version is A357637.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
Cf. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A156040
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Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.
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+10
19
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1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600
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OFFSET
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0,3
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COMMENTS
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For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957. - N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1 1 2 3 3 4 5 5 6 ...
1 1 2 3 3 4 5 ...
1 1 2 3 3 ...
1 1 2 ...
1 ...
------------------------------
a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two. - Gus Wiseman, Oct 09 2020
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LINKS
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FORMULA
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Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - N. J. A. Sloane, Apr 29 2011
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8. - Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, -1/3). - Heinrich Ludwig, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, -1]. - Michael Somos, Mar 26 2017
0 = a(n)*(-1 + a(n) - 2*a(n+1) - 2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2) - 2*a(n+3)) + a(n+2)*(+1 + a(n+2) - 2*a(n+3)) + a(n+3)*(-1 + a(n+3)) for all n in Z. - Michael Somos, Mar 26 2017
Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - Amiram Eldar, Dec 10 2022
E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - Stefano Spezia, Apr 05 2023
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EXAMPLE
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G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
(000) (100) (101) (111) (202) (212) (222) (313)
(110) (201) (211) (221) (303) (322)
(200) (210) (220) (302) (312) (331)
(300) (301) (311) (321) (403)
(310) (320) (330) (412)
(400) (401) (402) (421)
(410) (411) (430)
(500) (420) (502)
(501) (511)
(510) (520)
(600) (601)
(610)
(700)
(End)
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MAPLE
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a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009
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MATHEMATICA
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nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3, {3}], #[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]], {n, 0, 15}] (* Gus Wiseman, Oct 05 2020*)
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PROG
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CROSSREFS
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A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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