

A130665


a(n) = Sum_{k=0..n} 3^wt(k), where wt() = A000120().


26



1, 4, 7, 16, 19, 28, 37, 64, 67, 76, 85, 112, 121, 148, 175, 256, 259, 268, 277, 304, 313, 340, 367, 448, 457, 484, 511, 592, 619, 700, 781, 1024, 1027, 1036, 1045, 1072, 1081, 1108, 1135, 1216, 1225, 1252, 1279, 1360, 1387, 1468, 1549, 1792, 1801, 1828, 1855
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OFFSET

0,2


COMMENTS

Partial sums of A048883.  David Applegate, Jun 11 2009.
From Gary W. Adamson, Aug 26 2016: (Start)
The formula of Mar 26 2010 is equivalent to the leftshifted vector of matrix powers (Lim_{k=1..inf} M^k), of the production matrix M:
1, 0, 0, 0, 0, 0,...
4, 0, 0, 0, 0, 0,...
3, 1, 0, 0, 0, 0,...
0, 4, 0, 0, 0, 0,...
0, 3, 1, 0, 0, 0,...
0, 0, 4, 0, 0, 0,...
0, 0, 3, 1, 0, 0,...
... The sequence divided by its aerated variant is (1, 4, 3, 0, 0, 0,...). (End)


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
Tanya Khovanova and Joshua Xiong, Nim Fractals, arXiv:1405.594291 [math.CO], 2014 and J. Int. Seq. 17 (2014) # 14.7.8.
D. E. Knuth, Problem 11320 Amer. Math. Monthly, Vol. 114 No. 9 (Nov 2007) p. 835.
Omar E. Pol, Illustration of initial terms: Fig. 1. Neighbors of the vertices, Fig. 2. Overlapping squares, Fig. 3. Onestep bishop, (Nov 08 2009)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Mike Warburton, UlamWarburton Automaton  Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019.


FORMULA

With a different offset: a(1) = 1; a(n) = max { 3*a(k)+a(nk)  1 <= k <= n/2 }, for n>1.
a(2n+1) = 4a(n) and a(2n) = 3a(n1)+a(n).
a(n) = (A147562(n+1)1)*3/4 + 1.  Omar E. Pol, Nov 08 2009
a(n) = A160410(n+1)/4.  Omar E. Pol, Nov 12 2009
Let r(x) = (1 + 4x + 3x^2), then (1 + 4x + 7x^2 + 16x^3 + ...) =
r(x)* r(x^2) * r(x^4) * r(x^8) * ...  Gary W. Adamson, Mar 26 2010


MAPLE

u:=3; a[1]:=1; M:=30; for n from 1 to M do a[2*n] := (u+1)*a[n]; a[2*n+1] := u*a[n] + a[n+1]; od; t1:=[seq( a[n], n=1..2*M )]; # Gives sequence with a different offset


MATHEMATICA

f[n_] := Sum[3^Count[ IntegerDigits[k, 2], 1], {k, 0, n}]; Array[f, 51, 0] (* Robert G. Wilson v, Jun 28 2010 *)


PROG

(Haskell)
a130665 = sum . map (3 ^) . (`take` a000120_list) . (+ 1)
 Reinhard Zumkeller, Apr 18 2012


CROSSREFS

Cf. A006046, A116520, A130667, A147562, A151920, A151922, A160410, A160412.
Sequence in context: A266532 A160715 A160120 * A236305 A212062 A256926
Adjacent sequences: A130662 A130663 A130664 * A130666 A130667 A130668


KEYWORD

nonn,look


AUTHOR

N. J. A. Sloane, based on a message from Don Knuth, Jun 23 2007


EXTENSIONS

Simpler definition (and new offset) from David Applegate, Jun 11 2009
Lower limit of sum in definition changed from 1 to 0 by Robert G. Wilson v, Jun 28 2010


STATUS

approved



