A351837 Consider a variant of the toothpick sequence (A139250) where each new toothpick, except the first, touches exactly one existing toothpick at the ends, this one being in the prior stage; a(n) is the total number of toothpicks at stage n.

0, 1, 5, 9, 17, 25, 37, 53, 69, 77, 89, 109, 133, 161, 201, 249, 281, 289, 301, 321, 345, 373, 413, 465, 505, 533, 577, 641, 717, 813, 941, 1069, 1133, 1141, 1153, 1173, 1197, 1225, 1265, 1317, 1357, 1385, 1429, 1493, 1569, 1665, 1793, 1925, 1997, 2025, 2069

Offset: 0

Author

Rémy Sigrist, Feb 21 2022

Keywords

• nonn

Comments

We consider toothpicks of length 1, parallel to the X and Y axes.

We start at stage 0 with no toothpicks.

At stage 1 we place one toothpick anywhere in the plane.

At stage n > 1, we consider all exposed ends E (i.e. in contact with no other toothpick) and attach perpendicular toothpicks in contact with E by one end provided that they won't touch other existing toothpicks (from stages 1 to n-1).

A toothpick added at stage n may touch other toothpicks added at stage n.

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(4) = 1 + 4 + 4 + 8 = 17.
		

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. A139250, A351838.

Programs

PARI

See Links section.