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A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1. 10
1, 6, 11, 36, 41, 66, 91, 216, 221, 246, 271, 396, 421, 546, 671, 1296, 1301, 1326, 1351, 1476, 1501, 1626, 1751, 2376, 2401, 2526, 2651, 3276, 3401, 4026, 4651, 7776, 7781, 7806, 7831, 7956, 7981, 8106, 8231, 8856, 8881, 9006, 9131, 9756, 9881, 10506, 11131 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

The formula of Mar 26 2010 is equivalent to the following: Given the production matrix M below, Lim_{k=1..inf} M^k as a left-shifted vector generates the sequence.

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

6, 0, 0, 0, 0,...

5, 1, 0, 0, 0,...

0, 6, 0, 0, 0,...

0, 5, 1, 0, 0,...

0, 0, 6, 0, 0,...

0, 0, 5, 1, 0,...

...

The sequence divided by its aerated variant is (1, 6, 5, 0, 0, 0, ...). (End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

D. E. Knuth, Problem 11320, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.

FORMULA

a(2*n) = 6*a(n) and a(2*n+1) = 5*a(n) + a(n+1).

Let r(x) = (1 + 6*x + 5*x^2). Then (1 + 6*x + 11*x^2 + 36*x^3 + ...) = r(x) * r(x^2) * r(x^4) * r(x^8) * ... - Gary W. Adamson, Mar 26 2010

a(n) = Sum_{k=0..n} 5^wt(k), where wt = A000120. - Mike Warburton, Mar 14 2019

MAPLE

a:= proc(n) option remember;

      `if`(n=1, 1, `if`(irem(n, 2, 'm')=0, 6*a(m), 5*a(m)+a(n-m)))

    end:

seq(a(n), n=1..70); # Alois P. Heinz, Apr 09 2012

MATHEMATICA

a[1]=1; a[n_] := a[n] = If[EvenQ[n], 6a[n/2], 5a[(n-1)/2]+a[(n+1)/2]]; Array[a, 50] (* Jean-Fran├žois Alcover, Feb 13 2015 *)

PROG

(Haskell)

import Data.List (transpose)

a130667 n = a130667_list !! (n-1)

a130667_list = 1 : (concat $ transpose

   [zipWith (+) vs a130667_list, zipWith (+) vs $ tail a130667_list])

   where vs = map (* 5) a130667_list

-- Reinhard Zumkeller, Apr 18 2012

(PARI) first(n)=my(v=vector(n), r, t); v[1]=1; for(i=2, n, r=0; for(k=1, i\2, t=5*v[k]+v[i-k]; if(t>r, r=t)); v[i]=r); v \\ Charles R Greathouse IV, Aug 29 2016

(MAGMA) [&+[5^(2*k - Valuation(Factorial(2*k), 2)): k in [0..n]]: n in [0..50]]; // Vincenzo Librandi, Mar 15 2019

CROSSREFS

Cf. A000120, A006046, A116520, A130665.

Sequence in context: A100093 A219500 A166702 * A259669 A108698 A002570

Adjacent sequences:  A130664 A130665 A130666 * A130668 A130669 A130670

KEYWORD

nonn,changed

AUTHOR

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

STATUS

approved

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Last modified March 18 16:00 EDT 2019. Contains 321292 sequences. (Running on oeis4.)