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Partial sums of A160552.
10

%I #66 Feb 24 2021 02:48:19

%S 0,1,2,5,6,9,14,21,22,25,30,37,42,53,70,85,86,89,94,101,106,117,134,

%T 149,154,165,182,201,222,261,310,341,342,345,350,357,362,373,390,405,

%U 410,421,438,457,478,517,566,597,602,613,630,649,670,709,758,793,814,853,906,965,1046,1173,1302,1365,1366,1369,1374

%N Partial sums of A160552.

%C It appears that the sums of two successive terms give the positive terms of the toothpick sequence A139250.

%C It appears that the odd terms (a bisection) give A162795.

%C It appears that a(n) is also the total number of ON cells at stage n+1 in one of the four wedges of two-dimensional cellular automaton "Rule 750" using the von Neumann neighborhood (see A169707). Therefore a(n) is also the total number of ON cells at stage n+1 in one of the four quadrants of the NW-NE-SE-SW version of that cellular automaton.

%C See also the formula section.

%C First differs from A169779 at a(11).

%H Ivan Neretin, <a href="/A255747/b255747.txt">Table of n, a(n) for n = 0..8191</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.] <a href="http://arxiv.org/abs/1004.3036">arXiv:1004.3036</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.

%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>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F It appears that a(n) + a(n-1) = A139250(n), n >= 1.

%F It appears that a(2n-1) = A162795(n), n >= 1.

%F It appears that a(n) = (A169707(n+1) - 1)/4.

%e Also, written as an irregular triangle in which the row lengths are the terms of A011782 (the number of compositions of n, n >= 0), the sequence begins:

%e 0;

%e 1;

%e 2, 5;

%e 6, 9, 14, 21;

%e 22, 25, 30, 37, 42, 53, 70, 85;

%e 86, 89, 94,101,106,117,134,149,154,165,182,201,222,261,310,341;

%e ...

%e It appears that the first column gives 0 together with the terms of A047849, hence the right border gives A002450.

%e It appears that this triangle only shares with A151920 the positive elements of the columns 1, 2, 4, 8, 16, ... (the powers of 2).

%t Accumulate[Nest[Join[#, 2 # + Append[Rest@#, 1]] &, {0}, 6]] (* _Ivan Neretin_, Feb 09 2017 *)

%Y Cf. A002450, A011782, A047849, A139250, A151548, A151920, A160552, A162795, A169707, A169779, A246336.

%K nonn

%O 0,3

%A _Omar E. Pol_, Mar 05 2015