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%I #20 Jul 31 2023 17:35:26
%S 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,
%T 28,44,64,76,96,128,128,64,8,12,20,24,28,40,52,40,28,44,64,76,96,128,
%U 132,72,28,44,64,76,96,132,144,108,100,152,204,248,320,384,320
%N First differences of A351837.
%C Equivalently, a(n) gives the number of toothpicks added at stage n of the construction described in A351837.
%C For symmetry reasons, all terms except a(1) = 1 are multiples of 4.
%H Rémy Sigrist, <a href="/A351838/b351838.txt">Table of n, a(n) for n = 0..8194</a>
%H Rémy Sigrist, <a href="/A351837/a351837.png">Illustration of the structure at stage 16</a>
%H Rémy Sigrist, <a href="/A351838/a351838.gp.txt">PARI program</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F Empirically:
%F - a(2^k - 1) = A058922(k-1) for any k >= 2,
%F - a(2^k) = 2^(k+1) for any k >= 1,
%F - a(2^k + 1) = 8 for any k >= 2,
%F - a(2^k + 2) = 12 for any k >= 2.
%e The configuration at stage 4 can be depicted as follows (stars representing ends and toothpicks being labeled with their stage of appearance):
%e .
%e * *
%e | |
%e 4 4
%e | |
%e *---3---* *---3---*
%e | | | |
%e 4 2 2 4
%e | | | |
%e * *---1---* *
%e | | | |
%e 4 2 2 4
%e | | | |
%e *---3---* *---3---*
%e | |
%e 4 4
%e | |
%e * *
%e .
%e - so a(1) = 1, a(2) = a(3) = 4, a(4) = 8.
%o (PARI) See Links section.
%Y Cf. A058922, A139251, A351837.
%K nonn
%O 0,3
%A _Rémy Sigrist_, Feb 21 2022