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Search: a249095 -id:a249095
Displaying 1-6 of 6 results found.
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A249111
Triangle of partial sums of rows in triangle A249095.
+20
5
1, 1, 2, 3, 1, 2, 4, 5, 6, 1, 2, 5, 7, 10, 11, 12, 1, 2, 6, 9, 15, 18, 22, 23, 24, 1, 2, 7, 11, 21, 27, 37, 41, 46, 47, 48, 1, 2, 8, 13, 28, 38, 58, 68, 83, 88, 94, 95, 96, 1, 2, 9, 15, 36, 51, 86, 106, 141, 156, 177, 183, 190, 191, 192, 1, 2, 10, 17, 45, 66
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Length of row n = 2*n+1.
In the layout as given in the example, T(n,k) is the sum of the two elements to the left and to the right of the element just above, with the row continued to the left by 0's and to the right by the last element, cf. formula. - M. F. Hasler, Nov 17 2014
LINKS
Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
FORMULA
T(n,0) = A249095(n,0) = 1; T(n,k) = T(n,k-1) + A249095(n,k), k <= n.
T(n+1,k+1) = T(n,k-1) + T(n,k+1), with T(n,k-1)=0 for k<1 and T(n,k+1)=T(n,k) for k>=2n (last element of the row). In particular, T(n,k)=k+1 if k<2n and T(n,k)=3*2^(n-1) if k>=2n. - M. F. Hasler, Nov 17 2014
EXAMPLE
The triangle begins:
. 0: 1
. 1: 1 2 3
. 2: 1 2 4 5 6
. 3: 1 2 5 7 10 11 12
. 4: 1 2 6 9 15 18 22 23 24
. 5: 1 2 7 11 21 27 37 41 46 47 48
. 6: 1 2 8 13 28 38 58 68 83 88 94 95 96
. 7: 1 2 9 15 36 51 86 106 141 156 177 183 190 191 192
. 8: 1 2 10 17 45 66 122 157 227 262 318 339 367 374 382 383 384 .
It can be seen that the elements (except for row 1) are sum of the neighbors to the upper left and upper right, with the table continued to the left with 0's and to the right with the last = largest element of each row. E.g., 1=0+1, 2=0+2, 4=1+3, 5=2+3 (=1+4 in the next row), 6=3+3 (in row 2), 7=2+5 etc. - M. F. Hasler, Nov 17 2014
PROG
(Haskell)
a249111 n k = a249111_tabf !! n !! k
a249111_row n = a249111_tabf !! n
a249111_tabf = map (scanl1 (+)) a249095_tabf
(PARI) T(n, k)=if(k<2, k+1, if(k>=2*n-2, 3<<(n-1), T(n-1, k-2)+T(n-1, k))) \\ M. F. Hasler, Nov 17 2014
CROSSREFS
Cf. A005408 (row lengths), A128543 (row sums), A248574 (central terms), A008949.
KEYWORD
nonn,tabf
AUTHOR
Reinhard Zumkeller, Nov 14 2014
STATUS
approved
A249307
Triangle read by rows: A249095(n,k) * 2^k, k = 0 .. 2*n+1.
+20
5
1, 1, 2, 4, 1, 2, 8, 8, 16, 1, 2, 12, 16, 48, 32, 64, 1, 2, 16, 24, 96, 96, 256, 128, 256, 1, 2, 20, 32, 160, 192, 640, 512, 1280, 512, 1024, 1, 2, 24, 40, 240, 320, 1280, 1280, 3840, 2560, 6144, 2048, 4096, 1, 2, 28, 48, 336, 480, 2240, 2560, 8960, 7680
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Length of row n = 2*n+1;
T(n,2*n+1) = 4^n = A000302(n);
for n > 0: sum of row n = 7*5^(n-1), cf. A005055.
LINKS
Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
PROG
(Haskell)
a249307 n k = a249307_tabf !! n !! k
a249307_row n = a249307_tabf !! n
a249307_tabf = map (zipWith (*) a000079_list) a249095_tabf
CROSSREFS
Cf. A249095, A000079, A005408 (row lengths), A249308 (central terms), A000302 (right edge), A005055 (row sums, except for initial 1), A013609.
Cf. A013609.
KEYWORD
nonn,tabf
AUTHOR
Reinhard Zumkeller, Nov 14 2014
STATUS
approved
A128014
Central binomial coefficients C(2n,n) repeated.
+10
14
1, 1, 2, 2, 6, 6, 20, 20, 70, 70, 252, 252, 924, 924, 3432, 3432, 12870, 12870, 48620, 48620, 184756, 184756, 705432, 705432, 2704156, 2704156, 10400600, 10400600, 40116600, 40116600, 155117520, 155117520, 601080390, 601080390
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Binomial transform is A097893. Hankel transform is A128017.
Hankel transform of a(n+1) is A128018. - Paul Barry, Nov 23 2009
Number of 2n-bead balanced binary necklaces that are equivalent to their reverse. - Andrew Howroyd, Sep 29 2017
Number of ballot sequences of length n in which the vote is tied or decided by 1 vote. - Nachum Dershowitz, Aug 12 2020
Number of binary strings of length n that are abelian squares. - Michael S. Branicky, Dec 21 2020
LINKS
Table of n, a(n) for n=0..33.
FORMULA
G.f.: (1+x)/sqrt(1-4*x^2).
a(n) = C(n,n/2)*(1+(-1)^n)/2 + C(n-1,(n-1)/2)*(1-(-1)^n)/2.
a(n) = (1/Pi)*Integral_{x=-2..2} x^n*(1+x)/(x*sqrt(4-x^2)), as moment sequence.
E.g.f. of a(n+1): Bessel_I(0,2*x)+2*Bessel_I(1,2*x). - Paul Barry, Mar 26 2010
n*a(n) +(n-2)*a(n-1) +4*(-n+1)*a(n-2) +4*(-n+3)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) = 2^n*Product_{k=0..n-1} ((k/n+1/n)/2)^((-1)^k). - Peter Luschny, Dec 03 2013
From Reinhard Zumkeller, Nov 14 2014: (Start)
a(n) = A000984(floor(n/2)).
a(n) = A249095(n,n) = A249308(n) / 2^n. (End)
MATHEMATICA
(1+x)/Sqrt[1-4x^2] + O[x]^34 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 07 2017 *)
With[{cb=Table[Binomial[2n, n], {n, 0, 20}]}, Riffle[cb, cb]] (* Harvey P. Dale, Feb 17 2020 *)
PROG
(Haskell)
a128014 = a000984 . flip div 2
-- Reinhard Zumkeller, Nov 14 2014
CROSSREFS
Cf. A097893, A128017, A128018.
Cf. A000984, A249095, A249308.
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 11 2007
STATUS
approved
A105321
Convolution of binomial(1,n) and Gould's sequence A001316.
+10
10
1, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 34, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 36, 12, 16, 24, 24, 24, 32, 48, 40, 24, 32, 48, 48, 48, 64, 96, 66, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A universal function related to the spherical growth of repeated truncations of maps.
a(n) = (number of ones in row n of triangle A249133) = (number of odd terms in row n of triangle A249095) = A000120(A249184(n)). - Reinhard Zumkeller, Nov 14 2014
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Preprint series, Univ. of Ljubljana, Vol. 38 (2000), 696.
T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti. Sem. Mat. Fis. Univ. Modena, Vol. 49 (2001), 167-176.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS - Omar E. Pol, May 29 2010
FORMULA
G.f. (1+x)*Product{k>=0, 1+2x^(2^k)};
a(n) = Sum_{k=0..n, binomial(1, n-k)*Sum_{j=0..k, binomial(k, j) mod 2}}.
a(n) = 2*A048460(n) for n>=2. - Omar E. Pol, Jan 02 2011
a((2*n-1)*2^p) = (2^p+2)*A001316(n-1), p >= 0 and n >= 1, with a(0) = 1. - Johannes W. Meijer, Jan 28 2013
a(n) = A001316(n) + A001316(n-1) for n > 0. - Reinhard Zumkeller, Nov 14 2014
EXAMPLE
Contribution from Omar E. Pol, May 29 2010: (Start)
If written as a triangle:
1;
3;
4;
6,6;
6,8,12,10;
6,8,12,12,12,16,24,18;
6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,34;
6,8,12,12,12,16,24,20,12,16,24,24,24,32,48,36,12,16,24,24,24,32,48,40,24,32,48,48,48,64,96,66;
(End)
MAPLE
nmax := 74: A001316 := n -> if n <= -1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(log[2](nmax)) do for n from 1 to nmax/(p+2)+1 do a((2*n-1)*2^p) := (2^p+2) * A001316(n-1) od: od: a(0) :=1: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013
MATHEMATICA
f[n_] := Sum[Binomial[1, n - k]Mod[Binomial[k, j], 2], {k, 0, n}, {j, 0, k}]; Array[f, 75, 0] (* Robert G. Wilson v, Jun 28 2010 *)
PROG
(Haskell)
a105321 n = if n == 0 then 1 else a001316 n + a001316 (n - 1)
-- Reinhard Zumkeller, Nov 14 2014
(PARI) a(n) = sum(k=0, n, binomial(1, n-k)*sum(j=0, k, binomial(k, j) % 2)); \\ Michel Marcus, Apr 29 2018
CROSSREFS
Cf. A139250, A139251, A173522, A048460.
Cf. A001316, A000120, A249184, A249095, A249133.
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 01 2005
STATUS
approved
A249133
Triangle read by rows: interleaving successive pairs of rows of Sierpiński's triangle.
+10
7
1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0
COMMENTS
A105321(n) = number of ones in row n;
A249304(n) = number of zeros in row n;
numbers, when rows are concatenated: A249183, A249184.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10200
FORMULA
T(n,k) = A249095(n,k) mod 2.
EXAMPLE
. 0: 1
. 1: 1 1 1
. 2: 1 1 0 1 1
. 3: 1 1 1 0 1 1 1
. 4: 1 1 0 1 0 1 0 1 1
. 5: 1 1 1 0 0 0 0 0 1 1 1
. 6: 1 1 0 1 1 0 0 0 1 1 0 1 1
. 7: 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1
. 8: 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1
. 9: 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
. 10: 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1
. 11: 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1
. 12: 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1
. 13: 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1
. 14: 1 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 1
. 15: 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1
. 16: 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 .
MATHEMATICA
row[n_] := Mod[Riffle[Binomial[n, Range[0, n]], Binomial[n - 1, Range[0, n - 1]]], 2]; Table[row[n], {n, 0, 10}] // Flatten (* Amiram Eldar, Jul 28 2023 *)
PROG
(Haskell)
a249133 n k = a249133_tabf !! n !! k
a249133_row n = a249133_tabf !! n
a249133_tabf = map (map (flip mod 2)) a249095_tabf
CROSSREFS
Cf. A005408 (row lengths), A105321 (row sums), A249095, A249304, A249183, A249184, A047999 (Sierpiński).
KEYWORD
nonn,tabf
AUTHOR
Reinhard Zumkeller, Nov 14 2014
STATUS
approved
A249304
Number of zeros in row n of triangle A249133.
+10
6
0, 0, 1, 1, 3, 5, 5, 3, 7, 13, 13, 11, 13, 15, 13, 7, 15, 29, 29, 27, 29, 31, 29, 23, 29, 39, 37, 31, 33, 35, 29, 15, 31, 61, 61, 59, 61, 63, 61, 55, 61, 71, 69, 63, 65, 67, 61, 47, 61, 87, 85, 79, 81, 83, 77, 63, 73, 91, 85, 71, 73, 75, 61, 31, 63, 125, 125
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
a(n) = (number of even terms in row n of triangle A249095) = A023416(A249184(n)) = A055641(A249183(n)).
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = A048967(n) + A048967(n-1) for n > 0.
MATHEMATICA
Join[{0}, Plus @@@ Partition[Table[n + 1 - 2^DigitCount[n, 2, 1], {n, 0, 100}], 2, 1]] (* Amiram Eldar, Jul 28 2023 *)
PROG
(Haskell)
a249304 n = if n == 0 then 0 else a048967 n + a048967 (n - 1)
CROSSREFS
Cf. A249133, A048967, A249095, A023416, A055641, A249183, A249184.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 14 2014
STATUS
approved
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