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A351838 First differences of A351837.

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