Search: a070893 -id:a070893
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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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A000447
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a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.
(Formerly M4697 N2006)
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+10
84
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0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485
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OFFSET
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0,3
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COMMENTS
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4 times the variance of the area under an n-step random walk: e.g., with three steps, the area can be 9/2, 7/2, 3/2, 1/2, -1/2, -3/2, -7/2, or -9/2 each with probability 1/8, giving a variance of 35/4 or a(3)/4. - Henry Bottomley, Jul 14 2003
Number of standard tableaux of shape (2n-1,1,1,1) (n>=1). - Emeric Deutsch, May 30 2004
Also a(n) = (1/6)*(8*n^3-2*n), n>0: structured octagonal diamond numbers (vertex structure 9). Cf. A059722 = alternate vertex; A000447 = structured diamonds; and structured tetragonal anti-diamond numbers (vertex structure 9). Cf. A096000 = alternate vertex; A100188 = structured anti-diamonds. Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The n-th tetrahedral (or pyramidal) number is n(n+1)(n+2)/6. This sequence contains the tetrahedral numbers of A000292 obtained for n= 1,3,5,7,... (see A015219). - Valentin Bakoev, Mar 03 2009
Using three consecutive numbers u, v, w, (u+v+w)^3-(u^3+v^3+w^3) equals 18 times the numbers in this sequence. - J. M. Bergot, Aug 24 2011
Number of integer solutions to 1-n <= x <= y <= z <= n-1. - Michael Somos, Dec 27 2011
Also the number of cubes in the n-th Haüy square pyramid. - Eric W. Weisstein, Sep 27 2017
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REFERENCES
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G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955 [Annotated scans of just pages 742-743]
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
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FORMULA
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a(n) = binomial(2*n+1, 3) = A000292(2*n-1).
G.f.: x*(1+6*x+x^2)/(1-x)^4.
a(n) = -a(-n) for all n in Z.
a(0)=0, a(1)=1, a(2)=10, a(3)=35, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, May 25 2012
a(n) = v(n,n-1), where v(n,k) is the central factorial numbers of the first kind with odd indices. - Mircea Merca, Jan 25 2014
For any nonnegative integers m and n, 8*(n^3)*a(m) + 2*m*a(n) = a(2*m*n). - Ivan N. Ianakiev, Mar 04 2017
Sum_{n>=1} 1/a(n) = 6*log(2) - 3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 - 3*log(2). (End)
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EXAMPLE
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G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
a(0) = 0, which corresponds to the empty sum.
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MAPLE
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)
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PROG
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(PARI) {a(n) = n * (4*n^2 - 1) / 3};
(Haskell)
a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
(PARI) concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016
(Python)
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CROSSREFS
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(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
A000447 is related to partitions of 2^n into powers of 2, as it is shown in the formula, example and cross-references of A002577. - Valentin Bakoev, Mar 03 2009
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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Chrystal and Durell references from R. K. Guy, Apr 02 2004
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STATUS
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approved
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A006578
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Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).
(Formerly M3329)
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+10
37
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0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054
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OFFSET
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0,3
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COMMENTS
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Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009
We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010
Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010
Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation
a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003
a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014
Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022
E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023
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EXAMPLE
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G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...
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MAPLE
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with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008
# 2nd program
(6*n^2 + 4*n - 1 + (-1)^n)/8 ;
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MATHEMATICA
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Accumulate[LinearRecurrence[{1, 1, -1}, {0, 1, 3}, 100]] (* Harvey P. Dale, Sep 29 2013 *)
a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *)
a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 1, 4, 8}, 53] (* Ray Chandler, Aug 03 2015 *)
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PROG
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(PARI) {a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */
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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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EXTENSIONS
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STATUS
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approved
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A002717
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a(n) = floor(n(n+2)(2n+1)/8).
(Formerly M3827 N1569)
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+10
34
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0, 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411, 525, 658, 812, 988, 1188, 1413, 1665, 1945, 2255, 2596, 2970, 3378, 3822, 4303, 4823, 5383, 5985, 6630, 7320, 8056, 8840, 9673, 10557, 11493, 12483, 13528, 14630, 15790, 17010, 18291, 19635, 21043, 22517, 24058
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OFFSET
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0,3
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COMMENTS
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Number of triangles in triangular matchstick arrangement of side n, for n >= 1. Row sums of A085691.
We observe that the sequence is the transform of A006578 by the following transform T: T(u_0,u_1,u_2,u_3,...)=(u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In another terms v_p=sum(u_k,k=0..p) and the G.f phi_v of v is given by: phi_v=phi_u/(1-z). - Richard Choulet, Jan 28 2010
a(n) has the expansion (1*0)+(1*1)+(4*1)+(4*2)+(7*2)+(7*3)+... ,where the expansion stops when a(n) has n+1 number of terms. The expansion starts at (1*0), and progresses by alternating addition of 1 to the second number and 3 to the first number. - Arlu Genesis A. Padilla, Jun 04 2014
Taking the absolute values of each n-th difference and excluding the first n terms of each mentioned sequence, A002717 has the first difference A006578 (see formula of Michael Somos dated Jun 09 2014), the second difference A032766 (see 'partial sum' crossref), the third difference A000034, the fourth difference A000012, and the fifth to n-th difference A000004. - Arlu Genesis A. Padilla, Jun 12 2014
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REFERENCES
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J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = (1/16)*[2n(2n+1)(n+2)+cos(Pi*n)-1]. - Justin C. Bozonier (justinb67(AT)excite.com), Dec 05 2000
a(m+1)-2a(m)+2a(m-2)-a(m-3) = 3. - Len Smiley, Oct 08 2001
a(n) = (2n(2n+1)(n+2)+(-1)^n-1)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Oct 25 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*k*binomial(k+1,2).
G.f.: x(1+2x)/((1+x)(1-x)^4). - Simon Plouffe in his 1992 dissertation (with a different offset).
a(0)=0, a(1)=1, a(2)=5, a(3)=13, a(4)=27, a(n)=3*a(n-1)-2*a(n-2)-2*a(n-3)+ 3*a(n-4)- a(n-5). - Harvey P. Dale, Jan 20 2013
a(n) = Sum_{i=1..n} T(n-i+1)+T(n-2*i+1), where T(n)=n*(n+1)/2=A000217(n) if n>0 and 0 if n<=0. So we have a(n+2)-a(n)=(n+2)^2+(n+1)*(n+2)/2. - Maurice Mischler, Sep 08 2014
E.g.f.: (x*(2*x^2 + 11*x + 9)*cosh(x) + (2*x^3 + 11*x^2 + 9*x - 1)*sinh(x))/8. - Stefano Spezia, Jul 19 2022
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EXAMPLE
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f(3)=13 because the following figure contains 13 triangles if horizontal bars are added:
....... /\
...... /\/\
..... /\/\/\
G.f. = x + 5*x^2 + 13*x^3 + 27*x^4 + 48*x^5 + 78*x^6 + 118*x^7 + 170*x^8 + ...
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MAPLE
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MATHEMATICA
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Table[Floor[n(n+2)(2n+1)/8], {n, 0, 50}] (* or *) LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 5, 13, 27}, 50] (* Harvey P. Dale, Jan 20 2013 *)
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PROG
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(PARI) {a(n) = n * (n+2) * (2*n+1) \ 8};
(Magma) [Floor(n*(n+2)*(2*n+1)/8): n in [0..50]]; // Wesley Ivan Hurt, Jun 04 2014
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CROSSREFS
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Cf. A000292 number of triangles with same orientation as largest triangle, A002623 number of triangles pointing in opposite direction to largest triangle, A085691 number of triangles of side k in arrangement of side n.
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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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A194498
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T(n,k)=Number of ways to arrange k nonattacking queens on the lower triangle of an n X n board
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+10
6
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1, 0, 3, 0, 0, 6, 0, 0, 2, 10, 0, 0, 0, 12, 15, 0, 0, 0, 0, 38, 21, 0, 0, 0, 0, 12, 92, 28, 0, 0, 0, 0, 0, 82, 188, 36, 0, 0, 0, 0, 0, 8, 330, 344, 45, 0, 0, 0, 0, 0, 0, 118, 1008, 580, 55, 0, 0, 0, 0, 0, 0, 4, 802, 2566, 920, 66, 0, 0, 0, 0, 0, 0, 0, 114, 3708, 5742, 1390, 78, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,3
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COMMENTS
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Table starts
...1....0......0.......0........0........0........0........0........0......0..0
...3....0......0.......0........0........0........0........0........0......0..0
...6....2......0.......0........0........0........0........0........0......0..0
..10...12......0.......0........0........0........0........0........0......0..0
..15...38.....12.......0........0........0........0........0........0......0..0
..21...92.....82.......8........0........0........0........0........0......0..0
..28..188....330.....118........4........0........0........0........0......0..0
..36..344...1008.....802......114........0........0........0........0......0..0
..45..580...2566....3708.....1384.......64........0........0........0......0..0
..55..920...5742...13280.....9890.....1644.......10........0........0......0..0
..66.1390..11652...39734....50662....19306.....1210........0........0......0..0
..78.2020..21926..104000...205512...146718....27198......484........0......0..0
..91.2842..38802..244948...698688...820218...322782....26084.......38......0..0
.105.3892..65322..530632..2074530..3670288..2564988...531344....14870......0..0
.120.5208.105428.1072776..5525902.13846830.15372702..6285270...629414...3656..0
.136.6832.164214.2048056.13476246.45661556.74615814.51834064.11949712.501700.56
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LINKS
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EXAMPLE
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Some solutions for n=6 k=4
..0............0............0............0............0............0
..0.0..........0.0..........0.1..........0.0..........1.0..........0.1
..1.0.0........1.0.0........0.0.0........0.0.1........0.0.1........0.0.0
..0.0.1.0......0.0.0.1......0.0.1.0......1.0.0.0......0.0.0.0......1.0.0.0
..0.0.0.0.1....0.1.0.0.0....1.0.0.0.0....0.0.0.1.0....0.1.0.0.0....0.0.1.0.0
..0.1.0.0.0.0..0.0.0.0.1.0..0.0.0.1.0.0..0.1.0.0.0.0..0.0.0.1.0.0..0.0.0.0.1.0
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A082289
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Expansion of x^4*(2+x)/((1+x)*(1-x)^5).
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+10
4
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2, 9, 26, 59, 116, 206, 340, 530, 790, 1135, 1582, 2149, 2856, 3724, 4776, 6036, 7530, 9285, 11330, 13695, 16412, 19514, 23036, 27014, 31486, 36491, 42070, 48265, 55120, 62680, 70992, 80104, 90066, 100929, 112746, 125571, 139460, 154470
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OFFSET
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4,1
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LINKS
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FORMULA
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G.f.: x^4*(2+x)/((1+x)*(1-x)^5).
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) + 3. If sequence is also defined for n <= 3 by this equation, then a(n)=0 for 0 <= n <= 3 and a(n) = A070893(-n) for n < 0.
a(n) = (1/96)*(2*(n-2)*n*(3*n^2 - 10*n + 4) + 3*(-1)^n - 3). a(n) - a(n-2) = A006002(n-3) for n > 5. - Bruno Berselli, Aug 26 2011
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6); a(4)=2, a(5)=9, a(6)=26, a(7)=59, a(8)=116, a(9)=206. - Harvey P. Dale, Aug 26 2013
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MATHEMATICA
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Drop[CoefficientList[Series[x^4(2+x)/((1+x)(1-x)^5), {x, 0, 50}], x], 4] (* or *) LinearRecurrence[{4, -5, 0, 5, -4, 1}, {2, 9, 26, 59, 116, 206}, 50] (* Harvey P. Dale, Aug 26 2013 *)
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PROG
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(PARI) a(n)=polcoeff(if(n>0, x^4*(2+x)/((1+x)*(1-x)^5), x*(1+2*x)/((1+x)*(1-x)^5))+x*O(x^abs(n)), abs(n))
(Magma) [(1/96)*(2*(n-2)*n*(3*n^2-10*n+4)+3*(-1)^n-3): n in [4..50]]; // Vincenzo Librandi, Aug 29 2011
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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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A166278
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Square array A(n,k), n,k>=0, read by antidiagonals: A(n,k) is the total element sum of the k-fold f transform applied to the length n sequence of 1's. And f returns a sorted result after multiplying the elements in its input sequence with 1, 2, 3,... in descending size order.
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+10
3
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0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 6, 10, 10, 5, 0, 1, 8, 19, 20, 15, 6, 0, 1, 12, 33, 46, 35, 21, 7, 0, 1, 16, 63, 100, 94, 56, 28, 8, 0, 1, 24, 111, 220, 242, 172, 84, 36, 9, 0, 1, 32, 201, 488, 633, 514, 290, 120, 45, 10, 0, 1, 48, 369, 1104, 1643, 1518, 984, 460, 165, 55, 11
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OFFSET
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0,6
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LINKS
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EXAMPLE
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A(3,4) = 33, because f([1,1,1]) = [1,2,3], (f^2)([1,1,1]) = [3,3,4], (f^3)([1,1,1]) = [4,6,9], (f^4)([1,1,1]) = [9,12,12], and 9+12+12 = 33.
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
2, 3, 4, 6, 8, 12, ...
3, 6, 10, 19, 33, 63, ...
4, 10, 20, 46, 100, 220, ...
5, 15, 35, 94, 242, 633, ...
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MAPLE
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f:= l-> sort([seq(sort(l, `>`)[i]*i, i=1..nops(l))]):
A:= (n, k)-> add(i, i=(f@@k)([1$n])):
seq(seq(A(n, d-n), n=0..d), d=0..15);
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MATHEMATICA
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f[L_List] := f[L] = Sort[Reverse[Sort[L]]*Range[Length[L]]];
A[0, _] = 0; A[n_, 0] := n; A[n_, k_] := Total[Nest[f, Range[n], k-1]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A082290
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Expansion of (1+x+x^2)/((1+x^2)*(1+x)^4*(1-x)^5).
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+10
2
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1, 2, 6, 9, 19, 26, 46, 59, 94, 116, 172, 206, 290, 340, 460, 530, 695, 790, 1010, 1135, 1421, 1582, 1946, 2149, 2604, 2856, 3416, 3724, 4404, 4776, 5592, 6036, 7005, 7530, 8670, 9285, 10615, 11330, 12870, 13695, 15466, 16412, 18436, 19514, 21814, 23036
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OFFSET
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0,2
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1, 3, -3, -2, 2, -2, 2, 3, -3, -1, 1).
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FORMULA
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Euler transform of length 4 sequence [ 2, 3, -1, 1]. - Michael Somos, Feb 15 2006
G.f.: (1 + x + x^2) / ((1 + x^2) * (1 + x)^4 * (1 - x)^5).
a(n) = 3*a(n-2) - 2*a(n-4) - 2*a(n-6) + 3*a(n-8) - a(n-10) + 3.
a(n) = a(-9-n) for all n in Z.
a(n) = (6*n^4+108*n^3+666*n^2+1620*n+1251+(4*n^3+54*n^2+236*n+333)*(-1)^n-48*(-1)^((6*n-1+(-1)^n)/4))/1536. - Luce ETIENNE, Oct 23 2014
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EXAMPLE
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G.f. = 1 + 2*x + 6*x^2 + 9*x^3 + 19*x^4 + 26*x^5 + 46*x^6 + 59*x^7 + ...
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MATHEMATICA
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Table[(6 n^4 + 108 n^3 + 666 n^2 + 1620 n + 1251 + (4 n^3 + 54 n^2 + 236 n + 333) (-1)^n - 48 (-1)^((6 n - 1 + (-1)^n)/4))/1536, {n, 0, 50}] (* after Luce ETIENNE; or, by definition: *) CoefficientList[Series[(1 + x + x^2)/((1 + x^2)*(1 + x)^4*(1 - x)^5), {x, 0, 50}], x] (* Bruno Berselli, Oct 26 2014 *)
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PROG
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(PARI) {a(n) = if( n<-8, a(-9-n), polcoeff( (1 + x + x^2) / ((1 + x^2) *(1 + x)^4 * (1 - x)^5) + x * O(x^n), n))};
(Magma) [(6*n^4 +108*n^3 +666*n^2 +1620*n +1251 +(4*n^3 +54*n^2 +236*n +333)*(-1)^n -48*(-1)^Floor((6*n -1 +(-1)^n)/4))/1536: n in [0..50]]; // Vincenzo Librandi, Oct 23 2014
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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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A185958
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Accumulation array of the array max{n,k}, by antidiagonals.
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+10
2
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1, 3, 3, 6, 7, 6, 10, 13, 13, 10, 15, 21, 22, 21, 15, 21, 31, 34, 34, 31, 21, 28, 43, 49, 50, 49, 43, 28, 36, 57, 67, 70, 70, 67, 57, 36, 45, 73, 88, 94, 95, 94, 88, 73, 45, 55, 91, 112, 122, 125, 125, 122, 112, 91, 55, 66, 111, 139, 154, 160, 161, 160, 154, 139, 111, 66, 78, 133, 169, 190, 200, 203, 203, 200, 190, 169, 133, 78, 91, 157, 202, 230, 245, 251, 252, 251, 245, 230, 202, 157, 91, 105, 183, 238, 274, 295, 305, 308, 308, 305, 295, 274, 238, 183, 105
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OFFSET
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1,2
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COMMENTS
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A member of the accumulation chain
where A051125, written as a rectangular array M, is given by M(n,k)=max{n,k}. See A144112 for the definition of accumulation array.
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LINKS
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FORMULA
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O.g.f.: F(z,v) = -(v^2*z^3+v*z^3-3*v*z^2+1)/((v*z^2-v*z-z+1)^2*(v*z^2-1)*(z-1)*(v*z-1)).
T(n,k) = [v^k] 1/2*n^2*(v^(n+2)+1)/(1-v)^2+1/2*n*(3*v^(n+3)-7*v^(n+2)+7*v-3)/(-1+v)^3-1/2*v*((1-v^(1/2))^4*(-1)^n+(1+v^(1/2))^4)*v^(1/2*n)/(1-v)^4+(6*v^2+6*v^(n+2)+v^(n+4)-3*v^(n+3)-3*v+1)/(1-v)^4.
(End)
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EXAMPLE
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Northwest corner:
1....3....6....10....15
3....7....13...21....31
6....13...22...34....49
10...21...34...50....70
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MAPLE
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A := proc(n, k) option remember; ## n >= 0 and k = 0 .. n
if k < 0 or k > n then
0
elif n = 0 then
1
else
A(n-1, k) + A(n-1, k-1) - A(n-2, k-1) + max(n-k+1, k+1)
end if
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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