A130665 -id:A130665 - cp's OEIS frontend

A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

1, 6, 11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671, 1296, 1301, 1326, 1351, 1476, 1501, 1626, 1751, 2376, 2401, 2526, 2651, 3276, 3401, 4026, 4651, 7776, 7781, 7806, 7831, 7956, 7981, 8106, 8231, 8856, 8881, 9006, 9131, 9756, 9881, 10506, 11131

Offset: 1

N. J. A. Sloane, based on a message from Don Knuth, Jun 23 2007

nonn

From Gary W. Adamson, Aug 27 2016: (Start)
The formula of Mar 26 2010 is equivalent to the following: Given the production matrix M below, lim_{k->infinity} M^k as a left-shifted vector generates the sequence.
  1, 0, 0, 0, 0, ...
  6, 0, 0, 0, 0, ...
  5, 1, 0, 0, 0, ...
  0, 6, 0, 0, 0, ...
  0, 5, 1, 0, 0, ...
  0, 0, 6, 0, 0, ...
  0, 0, 5, 1, 0, ...
  ...
The sequence divided by its aerated variant is (1, 6, 5, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 1..16383
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, 32-33.
D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.

Cf. A000120, A006046, A116520, A130665, A360189.

import Data.List (transpose)
a130667 n = a130667_list !! (n-1)
a130667_list = 1 : (concat $ transpose
   [zipWith (+) vs a130667_list, zipWith (+) vs $ tail a130667_list])
   where vs = map (* 5) a130667_list
-- Reinhard Zumkeller, Apr 18 2012

[&+[5^(2*k - Valuation(Factorial(2*k), 2)): k in [0..n]]: n in [0..50]]; // Vincenzo Librandi, Mar 15 2019

a:= proc(n) option remember;
      `if`(n=1, 1, `if`(irem(n, 2, 'm')=0, 6*a(m), 5*a(m)+a(n-m)))
    end:
seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012

a[1]=1; a[n_] := a[n] = If[EvenQ[n], 6a[n/2], 5a[(n-1)/2]+a[(n+1)/2]]; Array[a, 50] (* Jean-François Alcover, Feb 13 2015 *)

first(n)=my(v=vector(n),r,t); v[1]=1; for(i=2,n, r=0; for(k=1,i\2, t=5*v[k]+v[i-k]; if(t>r, r=t)); v[i]=r); v \\ Charles R Greathouse IV, Aug 29 2016

a(2*n) = 6*a(n) and a(2*n+1) = 5*a(n) + a(n+1).
Let r(x) = (1 + 6*x + 5*x^2). Then (1 + 6*x + 11*x^2 + 36*x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Mar 26 2010
a(n) = Sum_{k=0..n} 5^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019
a(n) = Sum_{k=0..floor(log_2(n))} 5^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

A073121 a(n) = ra(ceiling(n/2)) + sa(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

Original entry on oeis.org

1, 4, 10, 16, 28, 40, 52, 64, 88, 112, 136, 160, 184, 208, 232, 256, 304, 352, 400, 448, 496, 544, 592, 640, 688, 736, 784, 832, 880, 928, 976, 1024, 1120, 1216, 1312, 1408, 1504, 1600, 1696, 1792, 1888, 1984, 2080, 2176, 2272, 2368, 2464, 2560, 2656, 2752

Offset: 1

Jeffrey Shallit, Aug 25 2002

nonn

A recurrence occurring in the analysis of a regular expression algorithm.

			a(1)=1, a(2) = 2*(a(1)+a(1)) = 4, a(3) = 2*(a(2)+a(1)) = 10.

K. Ellul, J. Shallit and M.-w. Wang, Regular expressions: new results and open problems, in Descriptional Complexity of Formal Systems (DCFS), Proceedings of workshop, London, Ontario, Canada, 21-24 August 2002, pp. 17-34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Jean-Paul Allouche and Jeffrey Shallit, The Ring of k-regular Sequences, II
F. Barbero, G. Gutin, M. Jones, and B. Sheng, Parameterized and approximation algorithms for the load coloring problem, Algorithmica 79, No. 1, 211-229 (2017). Prop 3.
Keh-Ning Chang and Shi-Chun Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Patrascu, Picture-Hanging Puzzles, arXiv:1203.3602 [cs.DS], 2012-2014.
Keith Ellul, Bryan Krawetz, Jeffrey Shallit, and Ming-wei Wang, Regular expressions: new results and open problems, Journal of Automata, Languages and Combinatorics, preprint.
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, 37.
Jean-Marc Luck, Revisiting log-periodic oscillations, arXiv:2403.00432 [cond-mat.stat-mech], 2024. See p. 14.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A053644, A053645, A254575.

Sequences of form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (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.

a073121 n = a053644 n * (fromIntegral n + 2 * a053645 n)
-- Reinhard Zumkeller, Mar 23 2012

a:= proc(n) option remember; `if`(n=1, 1,
      2*((m-> a(m)+a(n-m))(iquo(n, 2))))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Feb 01 2015

a[n_] := a[n] = If[n == 1, 1, 2*(a[Quotient[n, 2]] + a[n - Quotient[n, 2]])]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)
a[ n_] := If[ n < 1, 0, Module[{m = 1, A = 1}, While[m < n, m *= 2; A = (Normal[A] /. x -> x^2) 2 (1 + x)^2 - 1 + O[x]^m]; Coefficient[A, x, n - 1]]]; (* Michael Somos, Jul 04 2017 *)

{a(n) = n--; if( n<0, 0, my(m=1, A = 1 + O(x)); while(m<=n, m*=2; A = subst(A, x, x^2) * 2 * (1 + x)^2 - 1); polcoeff(A, n))}; /* Michael Somos, Jul 04 2017 */

a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c.
a(n) == 1 (mod 3), a(n+1)-a(n) = 3*A053644(n). If k >= 1: a(2^k)=4^k, a(3*2^k)=(10/9)*4^k. More generally a(m*2^k) = a(m)*4^k. Hence for any n, n^2 <= a(n) <= C*n^2 where C is a constant 1.125 < C < 1.14 and it seems that C = lim_{k->infinity} a(A001045(k))/A001045(k)^2 where A001045(k) =(2^n - (-1)^n)/3 is the Jacobsthal sequence. In other words, in the range 2^k <= n <= 2^(k+1) the maximum of a(n)/n^2 is reached for the only possible n in the Jacobsthal sequence. - Benoit Cloitre, Aug 26 2002
For any n, n^2 <= a(n) <= 9/8 * n^2. - Arnoud van der Leer, Sep 01 2019
a(n) = 2*(a(floor(n/2)) + a(ceiling(n/2))) for n >= 2; alternatively, a(n) = 2^c(n+2b) where n = 2^c + b, 0 <= b < 2^c
G.f.: 3*x/(1-x)^2 * ((2*x+1)/3 + Sum_{k>=1} 2^(k-1)*x^2^k). - Ralf Stephan, Apr 18 2003
G.f.: A(x) = 2 * (1/x + 2 + x) * A(x^2) - x. - Michael Somos, Jul 04 2017

Edited by N. J. A. Sloane, Feb 16 2016

A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

Original entry on oeis.org

0, 1, 8, 15, 64, 71, 120, 169, 512, 519, 568, 617, 960, 1009, 1352, 1695, 4096, 4103, 4152, 4201, 4544, 4593, 4936, 5279, 7680, 7729, 8072, 8415, 10816, 11159, 13560, 15961, 32768, 32775, 32824, 32873, 33216, 33265, 33608, 33951, 36352, 36401, 36744, 37087, 39488

Offset: 0

Omar E. Pol, Jun 14 2009

nonn

First differences are in A161343. - Omar E. Pol, May 03 2015
From Gary W. Adamson, Aug 30 2016: (Start)
Let M =
  1, 0, 0, 0, 0, ...
  8, 0, 0, 0, 0, ...
  7, 1, 0, 0, 0, ...
  0, 8, 0, 0, 0, ...
  0, 7, 1, 0, 0, ...
  0, 0, 8, 0, 0, ...
  0, 0, 7, 1, 0, ...
  ...
Then M^k converges to a single nonzero column giving the sequence.
The sequence with offset 1 divided by its aerated variant is (1, 8, 7, 0, 0, 0, ...). (End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A160410, A160428, A161343, A006046, A130665, A116520, A130667, A116522, A116526, A116525, A360189.

b:= proc(n) option remember; `if`(n<0, 0,
      b(n-1)+x^add(i, i=Bits[Split](n)))
    end:
a:= n-> subs(x=7, b(n-1)):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 06 2023

A161342list[nmax_]:=Join[{0},Accumulate[7^DigitCount[Range[0,nmax-1],2,1]]];A161342list[100] (* Paolo Xausa, Aug 05 2023 *)

From Nathaniel Johnston, Nov 13 2010: (Start)
a(n) = Sum_{k=0..n-1} 7^A000120(k).
a(n) = 1 + 7 * Sum_{k=1..n-1} A151785(k), for n >= 1.
a(2^n) = 2^(3n).
(End)
a(n) = Sum_{k=0..floor(log_2(n))} 7^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

More terms from Nathaniel Johnston, Nov 13 2010

A268524 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,1).

Original entry on oeis.org

1, 4, 13, 16, 43, 52, 61, 64, 145, 172, 199, 208, 235, 244, 253, 256, 499, 580, 661, 688, 769, 796, 823, 832, 913, 940, 967, 976, 1003, 1012, 1021, 1024, 1753, 1996, 2239, 2320, 2563, 2644, 2725, 2752, 2995, 3076, 3157, 3184, 3265, 3292, 3319, 3328, 3571, 3652, 3733, 3760, 3841, 3868, 3895

Offset: 1

N. J. A. Sloane, Feb 16 2016

nonn

Number of triples 0 <= i, j, k < n such that bitwise AND of all pairs (i, j), (j, k), (k, i) is 0. - Peter Karpov, Mar 01 2016

Start with A = [[[1]]], iteratively replace every element Aijk with Aijk * [[[1, 1], [1, 0]], [[1, 0], [0, 0]]]. a(n) is the sum of the resulting array inside the cubic region i, j, k < n. - Peter Karpov, Mar 01 2016

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, 3*a(ceil(n/2)) + a(floor(n/2))); \\ Michel Marcus, Mar 24 2016

A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647

Offset: 0

Omar E. Pol, May 20 2009, Jun 01 2009

nonn

From Omar E. Pol, Nov 10 2009: (Start)
On the infinite square grid, consider the outside corner of an infinite square.
We start at round 0 with all cells in the OFF state.
The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.
At round 2, we turn ON nine cells around the hexagon.
At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
And so on.
Shows a fractal-like behavior similar to the toothpick sequence A153006.
For the first differences see the entry A162349.
For more information see A160410, which is the main entry for this sequence.
(End)

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......

Michael De Vlieger, Table of n, a(n) for n = 0..1000
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.]
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, p. 31.
Omar E. Pol, Illustration of initial terms [From _Omar E. Pol_, Nov 10 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata - _Omar E. Pol_, Nov 10 2009

Cf. A139250, A139251, A153006, A152980, A160410, A160414.

Cf. A130665, A162349. - Omar E. Pol, Nov 10 2009

a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)

From Omar E. Pol, Nov 10 2009: (Start)
a(n) = A160410(n)*3/4.
a(0) = 0, a(n) = A130665(n-1)*3, for n>0.
(End)

More terms from Omar E. Pol, Nov 10 2009
Edited by Omar E. Pol, Nov 11 2009
More terms from Nathaniel Johnston, Nov 06 2010
More terms from Colin Barker, Apr 19 2015

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

Original entry on oeis.org

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

A173068 a(n) = A160120(n) - A160715(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 42, 42, 42, 42, 42, 42, 42, 42, 120

Offset: 0

Omar E. Pol, May 29 2010

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. A130665, A139250, A160120, A160715, A173066, A173067, A173451, A173452, A173453.

a(14)-a(33) from Robert Price, Jun 17 2019

cp's OEIS Frontend

A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

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A073121 a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

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A161342 Number of "ON" cubic cells at n-th stage in simple 3-dimensional cellular automaton: a(n) = A160428(n)/8.

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A268524 a(n) = r*a(ceiling(n/2))+s*a(floor(n/2)) with a(1)=1 and (r,s)=(3,1).

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A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

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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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A073121 a(n) = ra(ceiling(n/2)) + sa(floor(n/2)) with a(1)=1 and (r,s)=(2,2).

A268524 a(n) = ra(ceiling(n/2))+sa(floor(n/2)) with a(1)=1 and (r,s)=(3,1).

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,....