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%I #14 Feb 24 2021 02:48:18
%S 0,4,12,24,36,52,80,112,132,148,176,212,252,312,400,480,516,532,560,
%T 596,636,696,784,868,924,984,1076,1188,1328,1536,1792,1984,2052,2068,
%U 2096,2132,2172,2232,2320,2404,2460,2520,2612,2724,2864,3072,3328,3524,3612
%N a(n) = 4*A153006(n).
%C For the first differences see A147646. - _Omar E. Pol_, Jul 24 2009
%C It appears that a(n) is also the total path length of a toothpick structure as A139250 after n-th stage which is constructed following a special rule: toothpicks of the new generation have length 4 when are placed on the square grid (note that every toothpick has four components of length 1), but after every stage, one (or two) of the four components of every toothpick of the new generation is removed, if such component contains a endpoint of the toothpick and if such endpoint is touching the midpoint or the endpoint of another toothpick. The truncated endpoints of the toothpicks remain exposed forever. Note that there are three sizes of toothpicks in the structure: toothpicks of length 4, 3 and 2. a(n) is also the total number of components in the structure after n-th stage. a(n) is also the number of grid points that are covered after n-th stage, except the central point of the structure. The toothpick sequence A139250 gives the number of toothpicks after n-th stage. - _Omar E. Pol_, Oct 24 2011
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%Y Cf. A139250, A139251, A152980, A153006, A159793, A159794.
%Y Cf. A147646, A152980. - _Omar E. Pol_, Jul 24 2009
%K nonn
%O 0,2
%A _Omar E. Pol_, May 02 2009
%E More terms from _Omar E. Pol_, Jul 24 2009
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