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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.

%I A351837 #18 Jul 31 2023 17:35:18
%S A351837 0,1,5,9,17,25,37,53,69,77,89,109,133,161,201,249,281,289,301,321,345,
%T A351837 373,413,465,505,533,577,641,717,813,941,1069,1133,1141,1153,1173,
%U A351837 1197,1225,1265,1317,1357,1385,1429,1493,1569,1665,1793,1925,1997,2025,2069
%N 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.
%C A351837 We consider toothpicks of length 1, parallel to the X and Y axes.
%C A351837 We start at stage 0 with no toothpicks.
%C A351837 At stage 1 we place one toothpick anywhere in the plane.
%C A351837 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).
%C A351837 A toothpick added at stage n may touch other toothpicks added at stage n.
%H A351837 Rémy Sigrist, <a href="/A351837/b351837.txt">Table of n, a(n) for n = 0..8194</a>
%H A351837 Rémy Sigrist, <a href="/A351837/a351837.png">Illustration of the structure at stage 16</a>
%H A351837 Rémy Sigrist, <a href="/A351837/a351837.gp.txt">PARI program</a>
%H A351837 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A351837 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e A351837 The configuration at stage 4 can be depicted as follows (stars representing ends and toothpicks being labeled with their stage of appearance):
%e A351837 .
%e A351837          *                       *
%e A351837          |                       |
%e A351837          4                       4
%e A351837          |                       |
%e A351837          *---3---*       *---3---*
%e A351837          |       |       |       |
%e A351837          4       2       2       4
%e A351837          |       |       |       |
%e A351837          *       *---1---*       *
%e A351837          |       |       |       |
%e A351837          4       2       2       4
%e A351837          |       |       |       |
%e A351837          *---3---*       *---3---*
%e A351837          |                       |
%e A351837          4                       4
%e A351837          |                       |
%e A351837          *                       *
%e A351837 .
%e A351837 - so a(4) = 1 + 4 + 4 + 8 = 17.
%o A351837 (PARI) See Links section.
%Y A351837 Cf. A139250, A351838.
%K A351837 nonn
%O A351837 0,3
%A A351837 _Rémy Sigrist_, Feb 21 2022