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%I #16 Feb 24 2021 02:48:18
%S 0,1,2,4,5,7,10,13,14,16,19,22,25,30,36,40,41,43,46,49,52,57,63,67,70,
%T 75,81,87,95,106,116,121,122,124,127,130,133,138,144,148,151,156,162,
%U 168,176,187,197,202,205,210,216,222,230,241,251,258,266,277,289,303,322,343,358
%N Partial sums of A118977.
%C Can be considered toothpick sequence for N=1.
%C Based on a consistent set of rules for generating toothpick sequences.
%C Cf. A139250 where (1, 1, 3, 1, 3, 5, 7, ...) is convolved with (1, 2, 2, 2, ...) and A162958 where A162956 is convolved with (1, 3, 3, 3, ...); the present sequence can be considered toothpick N=1 since A118977 is convolved with (1, 1, 1, ...).
%C Arranged in array fashion, the first three toothpick sequences would be:
%C N=1: A163267: (1, 2, 4, 5, 7, 10, 13, 14, ...)
%C N=2: A139250: (1, 3, 7, 11, 15, 23, 35, 43, ...)
%C N=3: A162958: (1, 4, 10, 19, 25, 40, 67, 94, ...)
%C ...
%C Is there an illustration of this sequence using toothpicks? - _Omar E. Pol_, Dec 13 2016
%H G. C. Greubel, <a href="/A163267/b163267.txt">Table of n, a(n) for n = 0..1000</a>
%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>
%t a[0] = 0; a[1] = 1; a[n_] := a[n] = (j = n - 2^Floor[Log[2, n]]; a[j] + a[j + 1]); Table[Sum[a[n], {n, 0, k}], {k, 0, 20}] (* _G. C. Greubel_, Dec 12 2016 *)
%Y Cf. A118977, A139250, A162958.
%K nonn
%O 0,3
%A _Gary W. Adamson_, Jul 24 2009
%E Edited and extended by _N. J. A. Sloane_, Jan 07 2010
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