A116520 a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.

0, 1, 5, 9, 25, 29, 45, 61, 125, 129, 145, 161, 225, 241, 305, 369, 625, 629, 645, 661, 725, 741, 805, 869, 1125, 1141, 1205, 1269, 1525, 1589, 1845, 2101, 3125, 3129, 3145, 3161, 3225, 3241, 3305, 3369, 3625, 3641, 3705, 3769, 4025, 4089, 4345, 4601, 5625

Offset: 0

Author

Roger L. Bagula, Mar 15 2006

Keywords

• nonn
• look

Comments

Equivalently, a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(0)=0, a(1)=1, for (r,s) = (1,4). - N. J. A. Sloane, Feb 16 2016

A 5-divide version of A084230.

Zero together with the partial sums of A102376. - Omar E. Pol, May 05 2010

Also, total number of cubic ON cells after n generations in a three-dimensional cellular automaton in which A102376(n-1) gives the number of cubic ON cells in the n-th level of the structure starting from the top. An ON cell remains ON forever. The structure looks like an irregular stepped pyramid, with n >= 1. - Omar E. Pol, Feb 13 2015

From Gary W. Adamson, Aug 27 2016: (Start)

The formula of Mar 26 2010 is equivalent to lim_{k->infinity} M^k of the following production matrix M:

1, 0, 0, 0, 0, 0, ...

5, 0, 0, 0, 0, 0, ...

4, 1, 0, 0, 0, 0, ...

0, 5, 0, 0, 0, 0, ...

0, 4, 1, 0, 0, 0, ...

0, 0, 5, 0, 0, 0, ...

0, 0, 4, 1, 0, 0, ...

0, 0, 0, 5, 0, 0, ...

...

The sequence with offset 1 divided by its aerated variant is (1, 5, 4, 0, 0, 0, ...). (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• K.-N. Chang and S.-C. Tsai, Exact solution of a minimal recurrence, Inform. Process. Lett. 75 (2000), 61-64.
• H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 27, 32.
• D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A000120, A006046, A077465, A130665, A130667, A102376, A360189.

Sequences of the form a(n) = r*a(ceiling(n/2)) + s*a(floor(n/2)), a(1)=1, for (r,s) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), (1,4), (2,3), (3,2), (4,1): A000027, A006046, A064194, A130665, A073121, A268524, A116520, A268525, A268526, A268527.

Programs

Haskell

import Data.List (transpose)
a116520 n = a116520_list !! n
a116520_list = 0 : zs where
   zs = 1 : (concat $ transpose
                      [zipWith (+) vs zs, zipWith (+) vs $ tail zs])
      where vs = map (* 4) zs
-- Reinhard Zumkeller, Apr 18 2012

Maple

a:=proc(n) if n=0 then 0 elif n=1 then 1 elif n mod 2 = 0 then 5*a(n/2) else 4*a((n-1)/2)+a((n+1)/2) fi end: seq(a(n),n=0..52);

Mathematica

b[0] := 0 b[1] := 1 b[n_?EvenQ] := b[n] = 5*b[n/2] b[n_?OddQ] := b[n] = 4*b[(n - 1)/2] + b[(n + 1)/2] a = Table[b[n], {n, 1, 25}]

Formula

a(0) = 1, a(1) = 1; thereafter a(2n) = 5a(n) and a(2n+1) = 4a(n) + a(n+1).

Let r(x) = (1 + 5x + 4x^2). Then (1 + 5x + 9x^2 + 25x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... . - Gary W. Adamson, Mar 26 2010

a(n) = Sum_{k=0..n-1} 4^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

a(n) = Sum_{k=0..floor(log_2(n))} 4^k*A360189(n-1,k). - Alois P. Heinz, Mar 06 2023

Extensions

Edited by N. J. A. Sloane, Apr 16 2006, Jul 02 2008