A130665 -id:A130665 - cp's OEIS frontend

A116525 a(0)=1, a(1)=1, a(n) = 11a(n/2) for even n, and a(n) = 10a((n-1)/2) + a((n+1)/2) for odd n >= 3.

0, 1, 11, 21, 121, 131, 231, 331, 1331, 1341, 1441, 1541, 2541, 2641, 3641, 4641, 14641, 14651, 14751, 14851, 15851, 15951, 16951, 17951, 27951, 28051, 29051, 30051, 40051, 41051, 51051, 61051, 161051, 161061, 161161, 161261, 162261, 162361, 163361, 164361

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
    1,  0,  0, 0, 0, ...
   11,  0,  0, 0, 0, ...
   10,  1,  0, 0, 0, ...
    0, 11,  0, 0, 0, ...
    0, 10,  1, 0, 0, ...
    0,  0, 11, 0, 0, ...
    0,  0, 10, 1, 0, ...
  ...
Then lim_{k->infinity} M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 11, 10, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A084230.

Cf. A130665, A116520, A130667, A116522, A161342, A116526, A360189.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 11*a(n/2) else 10*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..42);

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 11*b[n/2]; b[n_?OddQ] := b[n] = 10*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

Let r(x) = (1 + 11x + 10x^2).  The sequence is r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Aug 30 2016
a(n) = Sum_{k=0..n-1} 10^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 10^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2005

A116526 a(0)=1, a(1)=1, a(n) = 9a(n/2) for even n >= 2, and a(n) = 8a((n-1)/2) + a((n+1)/2) for odd n >= 3.

Original entry on oeis.org

0, 1, 9, 17, 81, 89, 153, 217, 729, 737, 801, 865, 1377, 1441, 1953, 2465, 6561, 6569, 6633, 6697, 7209, 7273, 7785, 8297, 12393, 12457, 12969, 13481, 17577, 18089, 22185, 26281, 59049, 59057, 59121, 59185, 59697, 59761, 60273, 60785, 64881, 64945, 65457, 65969

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

A 9-divide version of A084230.
The interest this one has is in the prime form of even odd 2^n+1, 2^n.
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  9, 0, 0, 0, 0, ...
  8, 1, 0, 0, 0, ...
  0, 9, 0, 0, 0, ...
  0, 8, 1, 0, 0, ...
  0, 0, 9, 0, 0, ...
  0, 0, 8, 1, 0, ...
  ...
Then M^k converges to a single nonzero column giving the sequence.
The sequence divided by its aerated variant is (1, 9, 8, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 2501 terms from G. C. Greubel)
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 33.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A000120, A006046, A077465, A130665, A116520, A130667, A116522, A161342, A116525, A360189.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 9*a(n/2) else 8*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..45);

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 9*b[n/2]; b[n_?OddQ] := b[n] = 8*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

a(n) = Sum_{k=0..n-1} 8^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

a(n) = Sum_{k=0..floor(log_2(n))} 8^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Edited by N. J. A. Sloane, Apr 16 2006

A237686 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

Original entry on oeis.org

1, 7, 14, 50, 63, 105, 148, 364, 413, 491, 546, 798, 883, 1141, 1400, 2696, 2961, 3255, 3382, 3850, 3983, 4313, 4620, 6132, 6469, 6979, 7322, 8870, 9387, 10941, 12496, 20272, 21833, 23423, 23982, 25746, 26167, 26929, 27524, 30332, 30933

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

Partial sums of A237711.

			There is a position (0,0,0,0) with a total of zero. There are 6 positions with a total of 2 that are permutations of (0,0,1,1). Therefore, a(1)=7.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 16 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A237711 (first differences), A130665 (3 piles), A238147 (5 piles), A241522, A241718.

Table[Length[
  Select[Flatten[
    Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0,
      a}], 2], Total[#] <= a &]], {a, 0, 100, 2}]

a(2n+1) = 7a(n) + a(n-1), a(2n+2) = a(n+1) + 7a(n).

A256141 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, in which row n lists the partial sums of the n-th row of the square array of A256140.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 7, 9, 5, 1, 1, 6, 9, 16, 11, 6, 1, 1, 7, 11, 25, 19, 15, 7, 1, 1, 8, 13, 36, 29, 28, 19, 8, 1, 1, 9, 15, 49, 41, 45, 37, 27, 9, 1, 1, 10, 17, 64, 55, 66, 61, 64, 29, 10, 1, 1, 11, 19, 81, 71, 91, 91, 125, 67, 33, 11, 1, 1, 12, 21, 100, 89, 120, 127, 216, 129, 76, 37, 12, 1

Offset: 0

Omar E. Pol, Mar 16 2015

Questions:
Is also A130667 a row of this square array?
Is also A116522 a row of this square array?
Is also A116526 a row of this square array?
Is also A116525 a row of this square array?
Is also A116524 a row of this square array?

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
--------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A000027: 1, 2, 3,  4,  5,  6,  7,   8,   9,  10,  11,  12,  13,  14,  15
A006046: 1, 3, 5,  9, 11, 15, 19,  27,  29,  33,  37,  45,  49,  57,  65
A130665: 1, 4, 7, 16, 19, 28, 37,  64,  67,  76,  85, 112, 121, 148, 175
A116520: 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369
A130667? 1, 6,11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671
A116522? 1, 7,13, 49, 55, 91,127, 343, 349, 385, 421, 637, 673, 889,1105
A161342: 1, 8,15, 64, 71,120,169, 512, 519, 568, 617, 960,1009,1352,1695
A116526? 1, 9,17, 81, 89,153,217, 729, 737, 801, 865,1377,1441,1953,2465
.......: 1,10,19,100,109,190,271,1000,1009,1090,1171,1900,1981,2710,3439
A116525? 1,11,21,121,131,231,331,1331,1341,1441,1541,2541,2641,3641,4641
.......: 1,12,23,144,155,276,397,1728,1739,1860,1981,3312,3422,4764,6095

Index entries for sequences related to cellular automata

Cf. A000120, A255740, A255741, A256140.
First five rows are A000012, A000027, A006046, A130665, A116520. Row 7 is A161342.
First eight columns are A000012, A000027, A005408, A000290, A028387, A000384, A003215, A000578. Column 9 is A081437. Column 11 is A015237. Columns 13-15 are A005915, A005917, A000583.

A238147 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

Original entry on oeis.org

1, 11, 26, 126, 191, 341, 516, 1516, 2081, 2731, 3206, 4706, 5631, 7381, 9256, 19256, 24821, 30471, 33946, 40446, 44171, 48921, 52796, 67796, 76221, 85471, 91846, 109346, 119971, 138721, 158096, 258096, 313661, 369311

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

Partial sums of A238759.

			There is 1 position (0,0,0,0,0) with a total of zero. There are 10 positions with a total of 2 that are permutations of (0,0,0,1,1). Therefore, a(1)=11.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 18 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A238759 (first differences), A130665 (3 piles), A237686 (4 piles), A241523, A241731.

Table[Length[
  Select[Flatten[
    Table[{n, k, j, i, BitXor[n, k, j, i]}, {n, 0, a}, {k, 0, a}, {j,
      0, a}, {i, 0, a}], 3], #[[5]] <= a &]], {a, 0, 35}]
(* Second program: *)
a[n_] := a[n] = Which[n <= 1, {1, 11}[[n+1]], OddQ[n], 11 a[(n-1)/2] + 5 a[(n-1)/2 - 1], EvenQ[n], a[(n-2)/2 + 1] + 15*a[(n-2)/2]];
Array[a, 34, 0] (* Jean-François Alcover, Dec 14 2018 *)

a(2n+1) = 11a(n) + 5a(n-1), a(2n+2) = a(n+1) + 15a(n).

A160807 a(n) = A160799(n)/4.

Original entry on oeis.org

0, 1, 5, 12, 28, 47, 75, 112, 176, 243, 319, 404, 516, 637, 785, 960, 1216, 1475, 1743, 2020, 2324, 2637, 2977, 3344, 3792, 4249, 4733, 5244, 5836, 6455, 7155, 7936, 8960, 9987, 11023, 12068, 13140, 14221, 15329, 16464, 17680, 18905, 20157

Offset: 1

Omar E. Pol, Jun 14 2009

nonn

Cf. A160410, A161411, A160799.

Essentially partial sums of A130665.

a(n)=sum(i=0, n-1, (n-i)*3^hammingweight(i)) \\ Charles R Greathouse IV, Aug 25 2016

G.f.: x^2*Product_{i>=0} p(x^(2^i)) where p(x) = 1 + 5*x + 7*x^2 + 3*x^3. - Gary W. Adamson, Aug 25 2016 [edited by Jason Yuen, Oct 06 2024]

More terms from Max Alekseyev, Dec 12 2011

A267093 a(n) is the number of P-positions for n-modular Nim with 3 piles.

Original entry on oeis.org

1, 8, 7, 28, 19, 67, 37, 100, 67, 142, 85

Offset: 1

Tanya Khovanova and Karan Sarkar, Jan 10 2016

			P-positions of 2-modular Nim with 3 piles: (0,0,0), (1,1,1) and permutations of (0,1,2). Thus a(2) = 8.

Tanya Khovanova and Karan Sarkar, P-positions in Modular Extensions to Nim, arXiv:1508.07054 [math.CO], 2015.

Cf. A130665, A267092.

a(2n+1) = A130665(2n+1).

cp's OEIS Frontend

A116525 a(0)=1, a(1)=1, a(n) = 11*a(n/2) for even n, and a(n) = 10*a((n-1)/2) + a((n+1)/2) for odd n >= 3.

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A116526 a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3.

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A237686 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

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A256141 Square array read by antidiagonals upwards: T(n,k), n>=0, k>=0, in which row n lists the partial sums of the n-th row of the square array of A256140.

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A238147 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

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Mathematica

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A160807 a(n) = A160799(n)/4.

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A267093 a(n) is the number of P-positions for n-modular Nim with 3 piles.

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Author

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Examples

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Crossrefs

Formula

A116525 a(0)=1, a(1)=1, a(n) = 11a(n/2) for even n, and a(n) = 10a((n-1)/2) + a((n+1)/2) for odd n >= 3.

A116526 a(0)=1, a(1)=1, a(n) = 9a(n/2) for even n >= 2, and a(n) = 8a((n-1)/2) + a((n+1)/2) for odd n >= 3.