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%I #33 Nov 02 2022 07:44:54
%S 1,2,5,6,9,12,21,22,25,28,37,40,49,58,85,86,89,92,101,104,113,122,149,
%T 152,161,170,197,206,233,260,341,342,345,348,357,360,369,378,405,408,
%U 417,426,453,462,489,516,597,600,609,618,645,654,681,708,789,798,825,852,933,960
%N a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().
%C Partial sums of A147610 (but with offset changed to 0).
%C It appears that the first bisection gives the positive terms of A147562. - _Omar E. Pol_, Mar 07 2015
%H Michael De Vlieger, <a href="/A151920/b151920.txt">Table of n, a(n) for n = 0..10000</a>
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 31.
%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 a(n) = (A147562(n+2) - 1)/4 = (A151917(n+2) - 1)/2. - _Omar E. Pol_, Mar 13 2011
%F a(n) = (A130665(n+1) - 1)/3. - _Omar E. Pol_, Mar 07 2015
%F a(n) = a(n-1) + 3^A000120(n+1))/3. - _David A. Corneth_, Mar 21 2015
%e n=3: (3^1+3^1+3^2+3^1)/3 = 18/3 = 6.
%e n=18: the binary expansion of 18+1 is 10011, i.e., 19 = 2^4 + 2^1 + 2^0.
%e The exponents of these powers of 2 (4, 1 and 0) reoccur as exponents in the powers of 4: a(19) = 3^0 * [(4^4 - 1) / 3 + 1] + 3^1 * [(4^1 - 1) / 3 + 1] + 3^2 * [(4^0 - 1)/3 + 1] = 1 * 86 + 3 * 2 + 9 * 1 = 101. - _David A. Corneth_, Mar 21 2015
%t t = Nest[Join[#, # + 1] &, {0}, 14]; Table[Sum[3^t[[i + 1]], {i, 1, n}]/3, {n, 60}] (* _Michael De Vlieger_, Mar 21 2015 *)
%o (PARI) a(n) = sum(i=1, n+1, 3^hammingweight(i))/3; \\ _Michel Marcus_, Mar 07 2015
%o (PARI) a(n)={b=binary(n+1);t=#b;e=-1;sum(i=1,#b,e+=(b[i]==1);(b[i]==1)*3^e*((4^(#b-i)-1)/3+1))} \\ _David A. Corneth_, Mar 21 2015
%Y Cf. A000120, A130665, A147562, A147610, A153000, A160410, A160414.
%Y Cf. A139250, A151917, A152998.
%K nonn,easy,look
%O 0,2
%A _N. J. A. Sloane_, Aug 05 2009, Aug 06 2009
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