A116520 -id:A116520 - cp's OEIS frontend

A268525 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(2,3).

1, 5, 13, 25, 41, 65, 89, 125, 157, 205, 253, 325, 373, 445, 517, 625, 689, 785, 881, 1025, 1121, 1265, 1409, 1625, 1721, 1865, 2009, 2225, 2369, 2585, 2801, 3125, 3253, 3445, 3637, 3925, 4117, 4405, 4693, 5125, 5317, 5605, 5893, 6325, 6613, 7045, 7477, 8125, 8317, 8605, 8893, 9325, 9613, 10045

Offset: 1

N. J. A. Sloane, Feb 16 2016

nonn

K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.

Sequences of form a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

[n le 1 select 1 else 2*Self(Ceiling(n/2))+3*Self(Floor(n/2)): n in [1..60]]; // Vincenzo Librandi, Aug 30 2016

a(n) = if (n==1, 1, 2*a(ceil(n/2))+3*a(floor(n/2))); \\ Michel Marcus, Aug 30 2016

A268526 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,2).

Original entry on oeis.org

1, 5, 17, 25, 61, 85, 109, 125, 233, 305, 377, 425, 497, 545, 593, 625, 949, 1165, 1381, 1525, 1741, 1885, 2029, 2125, 2341, 2485, 2629, 2725, 2869, 2965, 3061, 3125, 4097, 4745, 5393, 5825, 6473, 6905, 7337, 7625, 8273, 8705, 9137, 9425, 9857, 10145, 10433, 10625, 11273, 11705, 12137, 12425

Offset: 1

N. J. A. Sloane, Feb 16 2016

nonn

K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.

Sequences of form a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

[n le 1 select 1 else 3*Self(Ceiling(n/2))+2*Self(Floor(n/2)): n in [1..60]]; // Vincenzo Librandi, Aug 30 2016

a(n) = if (n==1, 1, 3*a(ceil(n/2))+2*a(floor(n/2))); \\ Michel Marcus, Aug 30 2016

A268527 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(4,1).

Original entry on oeis.org

1, 5, 21, 25, 89, 105, 121, 125, 381, 445, 509, 525, 589, 605, 621, 625, 1649, 1905, 2161, 2225, 2481, 2545, 2609, 2625, 2881, 2945, 3009, 3025, 3089, 3105, 3121, 3125, 7221, 8245, 9269, 9525, 10549, 10805, 11061, 11125, 12149, 12405, 12661, 12725, 12981, 13045, 13109, 13125, 14149, 14405

Offset: 1

N. J. A. Sloane, Feb 16 2016

nonn

K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.

Sequences of form a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

a(n) = if (n==1, 1, 4*a(ceil(n/2))+a(floor(n/2))); \\ Michel Marcus, Aug 30 2016

A116522 a(0)=1, a(1)=1, a(n)=7a(n/2) for n=2,4,6,..., a(n)=6a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

Original entry on oeis.org

0, 1, 7, 13, 49, 55, 91, 127, 343, 349, 385, 421, 637, 673, 889, 1105, 2401, 2407, 2443, 2479, 2695, 2731, 2947, 3163, 4459, 4495, 4711, 4927, 6223, 6439, 7735, 9031, 16807, 16813, 16849, 16885, 17101, 17137, 17353, 17569, 18865, 18901, 19117, 19333

Offset: 0

Roger L. Bagula, Mar 15 2006

nonn

A 7-divide version of A084230.
The Harborth: f(2^k) = 3^k suggests that a family of sequences of the form: f(2^k) = prime(n)^k.
From Gary W. Adamson, Aug 27 2016: (Start)
Let M = the production matrix below. Then lim_{k->infinity} M^k generates the sequence with offset 1 by extracting the left-shifted vector.
  1, 0, 0, 0, 0, ...
  7, 0, 0, 0, 0, ...
  6, 1, 0, 0, 0, ...
  0, 7, 0, 0, 0, ...
  0, 6, 1, 0, 0, ...
  0, 0, 7, 0, 0, ...
  0, 0, 6, 1, 0, ...
  ...
The sequence divided by its aerated variant is (1, 7, 6, 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, A161342, A360189.

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 7*a(n/2) else 6*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..47);
# second Maple program:
b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=6, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

b[0] := 0; b[1] := 1; b[n_?EvenQ] := b[n] = 7*b[n/2]; b[n_?OddQ] := b[n] = 6*b[(n - 1)/2] + b[(n + 1)/2]; a = Table[b[n], {n, 1, 25}]

G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + 7x + 6x^2).
a(n) = Sum_{k=0..n-1} 6^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 6^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

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

A253908 Partial sums of A072272.

Original entry on oeis.org

1, 6, 11, 28, 33, 58, 75, 136, 141, 166, 191, 276, 293, 378, 439, 656, 661, 686, 711, 796, 821, 946, 1031, 1336, 1353, 1438, 1523, 1812, 1873, 2178, 2395, 3168, 3173, 3198, 3223, 3308, 3333, 3458, 3543, 3848, 3873, 3998, 4123, 4548, 4633, 5058, 5363, 6448, 6465, 6550, 6635, 6924, 7009, 7434, 7723, 8760, 8821, 9126, 9431, 10468

Offset: 0

Omar E. Pol, Jan 27 2015

nonn

Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton where A072272(n) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid.

N. J. A. Sloane, Table of n, a(n) for n = 0..8192
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A072272, A116520, A245542, A253767.

Offset changed to 0 by N. J. A. Sloane, Feb 06 2015

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.

Original entry on oeis.org

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

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.

A253767 Partial sums of A247666.

Original entry on oeis.org

1, 8, 15, 40, 47, 96, 121, 224, 231, 280, 329, 504, 529, 704, 807, 1216, 1223, 1272, 1321, 1496, 1545, 1888, 2063, 2784, 2809, 2984, 3159, 3784, 3887, 4608, 5017, 6656, 6663, 6712, 6761, 6936, 6985, 7328, 7503, 8224, 8273, 8616, 8959, 10184, 10359, 11584, 12305, 15168, 15193, 15368, 15543, 16168

Offset: 0

Omar E. Pol, Jan 29 2015

nonn

Also, total number of ON cells after n generations in a three-dimensional cellular automaton where A247666(n) gives the number of ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. An ON cell is an hexagonal prism of height 1. We start with a single ON cell. The structure looks like an irregular stepped pyramid, apparently with a like-hexagonal base.

Cf. A116520, A245542, A247666, A253908.

A177240 Number of K-toothpicks after n stages of 3-D K-toothpick structure defined in Comments.

Original entry on oeis.org

0, 1, 5, 9, 25, 29, 45

Offset: 0

Omar E. Pol, May 05 2010

We are in 3-D. Here the polytoothpick is a K-toothpick. The K-toothpick has 4 components or line segments, a central point and 4 endpoints, as a tetrapod but without volume. The K-toothpick endpoints coincide with the vertices of a regular tetrahedron.
It appears that this is a three-dimensional version of A160120, but with K-toothpicks, not with Y-toothpick.
The first differences are in the entry A177241.
For the toothpick mechanism see A139250 and A160120.
Question: Is this the same as A116520? (To answer the question we need a program because the structure is hard to visualize).

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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A102376, A116520, A139250, A160120, A177241.

Edited by Omar E. Pol, May 07 2010

cp's OEIS Frontend

A268525 a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s)=(2,3).

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A268526 a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s)=(3,2).

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A268527 a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s)=(4,1).

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A116522 a(0)=1, a(1)=1, a(n)=7*a(n/2) for n=2,4,6,..., a(n)=6*a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

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A253908 Partial sums of A072272.

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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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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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A253767 Partial sums of A247666.

A268525 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(2,3).

A268526 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,2).

A268527 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(4,1).

A116522 a(0)=1, a(1)=1, a(n)=7a(n/2) for n=2,4,6,..., a(n)=6a((n-1)/2)+a((n+1)/2) for n=3,5,7,....

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.