A351837 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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.

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

EXAMPLE

The configuration at stage 4 can be depicted as follows (stars representing ends and toothpicks being labeled with their stage of appearance):

* *

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4 4