A323651 Number of elements added at n-th stage to the toothpick structure of A323650.

1, 2, 4, 8, 4, 8, 12, 24, 4, 8, 12, 24, 12, 24, 36, 72, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216, 12, 24, 36, 72, 36, 72, 108, 216, 36, 72, 108, 216, 108, 216, 324, 648, 4, 8, 12, 24, 12, 24, 36, 72, 12, 24, 36, 72, 36, 72, 108, 216

Offset

1,2

Comments

The odd-indexed terms (a bisection) gives A147582, the first differences of A147562 (Ulam-Warburton cellular automaton).

The even-indexed terms (a bisection) gives A147582 multiplied by 2.

The word of this cellular automaton is "ab", so the structure of the irregular triangle is as shown below:

a,b;

a,b;

a,b,a,b;

a,b,a,b,a,b,a,b;

a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;

...

Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.

Columns "a" contain numbers of I-toothpicks. Columns "b" contain numbers of V-toothpicks. See the example.

For further information about the word of cellular automata see A296612.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

Index entries for sequences related to cellular automata

Index entries for sequences related to toothpick sequences

Formula

a(2*n-1) = A147582(n).

a(2*n) = 2*A147582(n).

a(n) = 4*A323641(n-2), n >= 3.

Example

Written as an irregular triangle the sequence begins:
1,2;
4,8;
4,8,12,24;
4,8,12,24,12,24,36,72;
4,8,12,24,12,24,36,72,12,24,36,72,36,72,108,216;
4,8,12,24,12,24,36,72,12,24,36,72,36,72,108,216,12,24,36,72,36,72,108,216,...
...

Mathematica

Riffle[#, 2*#] & [Join[{1}, 4*3^(DigitCount[Range[50], 2, 1] - 1)]] (* Paolo Xausa, Dec 18 2025 *)

Crossrefs

First differences of A323650.

Cf. A011782, A139250, A139251, A147562, A147582, A160120, A160121, A161206, A161207, A296612, A323641, A323642, A323649.

For other hybrid cellular automata, see A194701, A194271, A220501, A290221, A294021, A294981.

Sequence in context: A078479 A060968 A384715 * A151569 A016635 A133992

Adjacent sequences: A323648 A323649 A323650 * A323652 A323653 A323654

Keyword

nonn

Author

Omar E. Pol, Feb 04 2019

Status

approved