A151780 -id:A151780 - cp's OEIS frontend

A151779 a(1)=1; for n > 1, a(n)=6*5^{wt(n-1)-1}.

1, 6, 6, 30, 6, 30, 30, 150, 6, 30, 30, 150, 30, 150, 150, 750, 6, 30, 30, 150, 30, 150, 150, 750, 30, 150, 150, 750, 150, 750, 750, 3750, 6, 30, 30, 150, 30, 150, 150, 750, 30, 150, 150, 750, 150, 750, 750, 3750, 30, 150, 150, 750, 150, 750, 750, 3750, 150, 750, 750, 3750

Offset: 1

N. J. A. Sloane, Jun 25 2009

Number of cells turned ON in n-th generation of cellular automaton based on Z^3 lattice in the same way that A147562 is based on the Z^2 lattice. Here each cell has six neighbors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A147582, A151780, A151781, A151782.

wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=d->[seq((2*d)*(2*d-1)^(wt(n-1)-1),n=2..120)];
f2:=d->[1,op(f(d))];
f2(3);

a(n)=6*5^(hammingweight(n-1)-1)\1 \\ Charles R Greathouse IV, Mar 07 2015

A151782 a(1)=1; for n > 1, a(n)=8*7^{wt(n-1)-1}.

Original entry on oeis.org

1, 8, 8, 56, 8, 56, 56, 392, 8, 56, 56, 392, 56, 392, 392, 2744, 8, 56, 56, 392, 56, 392, 392, 2744, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 8, 56, 56, 392, 56, 392, 392, 2744, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

Cf. A151779, A151780, etc.

A151783 a(n) = 4^(wt(n) - 1) where wt(n) = A000120(n).

Original entry on oeis.org

1, 1, 4, 1, 4, 4, 16, 1, 4, 4, 16, 4, 16, 16, 64, 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,4;
  1,4,4,16;
  1,4,4,16,4,16,16,64;
  1,4,4,16,4,16,16,64,4,16,16,64,16,64,64,256;
  1,4,4,16,4,16,16,64,4,16,16,64,16,64,64,256,4,16,16,64,16,64,64,256,16,64,...
(End)

Cf. A000120, A048881, A048896, A147610, A151780.
Cf. A000302, A102376. [Omar E. Pol, Jul 21 2009]
A102376 is a very similar sequence.

A256135 a(n) = 5^A000120(n).

Original entry on oeis.org

1, 5, 5, 25, 5, 25, 25, 125, 5, 25, 25, 125, 25, 125, 125, 625, 5, 25, 25, 125, 25, 125, 125, 625, 25, 125, 125, 625, 125, 625, 625, 3125, 5, 25, 25, 125, 25, 125, 125, 625, 25, 125, 125, 625, 125, 625, 625, 3125, 25, 125, 125, 625, 125, 625, 625, 3125, 125, 625, 625, 3125, 625, 3125, 3125, 15625

Offset: 0

Omar E. Pol, Mar 19 2015

nonn

Also, a row of the square array A256140.

It appears that when A151780 is regarded as a triangle in which the row lengths are the powers of 2, this is what the rows converge to.

			Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
5;
5, 25;
5, 25, 25, 125;
5, 25, 25, 125, 25, 125, 125, 625;
...

Cf. A000120, A001316, A048883, A102376, A130667, A151780, A256140, A256141.

5^# & /@ Nest[Join[#, # + 1] &, {0}, 6] (* Michael De Vlieger, Mar 20 2015, after IWABUCHI Yu(u)ki at a000120 *)

a(n) = 5^hammingweight(n); \\ Michel Marcus, Mar 21 2015

a(n) = A000351(A000120(n)). - Michel Marcus, Mar 21 2015

G.f.: Product_{k>=0} (1 + 5*x^(2^k)). - Ilya Gutkovskiy, Feb 28 2017

More terms from Michael De Vlieger, Mar 20 2015

A151784 a(n) = 6^(wt(n) - 1) where wt(n) = A000120(n).

Original entry on oeis.org

1, 1, 6, 1, 6, 6, 36, 1, 6, 6, 36, 6, 36, 36, 216, 1, 6, 6, 36, 6, 36, 36, 216, 6, 36, 36, 216, 36, 216, 216, 1296, 1, 6, 6, 36, 6, 36, 36, 216, 6, 36, 36, 216, 36, 216, 216, 1296, 6, 36, 36, 216, 36, 216, 216, 1296, 36, 216, 216, 1296, 216, 1296, 1296, 7776, 1, 6, 6, 36, 6, 36, 36

Offset: 1

N. J. A. Sloane, Jun 25 2009

nonn

			From _Omar E. Pol_, Jul 21 2009: (Start)
If written as a triangle:
  1;
  1,6;
  1,6,6,36;
  1,6,6,36,6,36,36,216;
  1,6,6,36,6,36,36,216,6,36,36,216,36,216,216,1296;
  1,6,6,36,6,36,36,216,6,36,36,216,36,216,216,1296,6,36,36,216,36,216,216,...
(End)

Cf. A000120, A048881, A048896, A147610, A151780, A151783, A000120.

Cf. A000400. - Omar E. Pol, Jul 21 2009

a(n) = 6^(hammingweight(n)-1); \\ Michel Marcus, Nov 15 2022

cp's OEIS Frontend

A151779 a(1)=1; for n > 1, a(n)=6*5^{wt(n-1)-1}.

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A151782 a(1)=1; for n > 1, a(n)=8*7^{wt(n-1)-1}.

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A151783 a(n) = 4^(wt(n) - 1) where wt(n) = A000120(n).

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A256135 a(n) = 5^A000120(n).

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A151784 a(n) = 6^(wt(n) - 1) where wt(n) = A000120(n).

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