A351838 First differences of A351837.

0, 1, 4, 4, 8, 8, 12, 16, 16, 8, 12, 20, 24, 28, 40, 48, 32, 8, 12, 20, 24, 28, 40, 52, 40, 28, 44, 64, 76, 96, 128, 128, 64, 8, 12, 20, 24, 28, 40, 52, 40, 28, 44, 64, 76, 96, 128, 132, 72, 28, 44, 64, 76, 96, 132, 144, 108, 100, 152, 204, 248, 320, 384, 320

Offset: 0

Author

Rémy Sigrist, Feb 21 2022

Keywords

• nonn

Comments

Equivalently, a(n) gives the number of toothpicks added at stage n of the construction described in A351837.

For symmetry reasons, all terms except a(1) = 1 are multiples of 4.

Examples

			The configuration at stage 4 can be depicted as follows (stars representing ends and toothpicks being labeled with their stage of appearance):
.
         *                       *
         |                       |
         4                       4
         |                       |
         *---3---*       *---3---*
         |       |       |       |
         4       2       2       4
         |       |       |       |
         *       *---1---*       *
         |       |       |       |
         4       2       2       4
         |       |       |       |
         *---3---*       *---3---*
         |                       |
         4                       4
         |                       |
         *                       *
.
- so a(1) = 1, a(2) = a(3) = 4, a(4) = 8.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8194
• Rémy Sigrist, Illustration of the structure at stage 16
• Rémy Sigrist, PARI program
• Index entries for sequences related to cellular automata
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A058922, A139251, A351837.

Programs

PARI

See Links section.

Formula

Empirically:

- a(2^k - 1) = A058922(k-1) for any k >= 2,

- a(2^k) = 2^(k+1) for any k >= 1,

- a(2^k + 1) = 8 for any k >= 2,

- a(2^k + 2) = 12 for any k >= 2.