cp's OEIS Frontend

A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.

%I A130667 #64 Mar 06 2023 17:25:18
%S A130667 1,6,11,36,41,66,91,216,221,246,271,396,421,546,671,1296,1301,1326,
%T A130667 1351,1476,1501,1626,1751,2376,2401,2526,2651,3276,3401,4026,4651,
%U A130667 7776,7781,7806,7831,7956,7981,8106,8231,8856,8881,9006,9131,9756,9881,10506,11131
%N A130667 a(1) = 1; a(n) = max{ 5*a(k) + a(n-k) | 1 <= k <= n/2 } for n > 1.
%C A130667 From _Gary W. Adamson_, Aug 27 2016: (Start)
%C A130667 The formula of Mar 26 2010 is equivalent to the following: Given the production matrix M below, lim_{k->infinity} M^k as a left-shifted vector generates the sequence.
%C A130667   1, 0, 0, 0, 0, ...
%C A130667   6, 0, 0, 0, 0, ...
%C A130667   5, 1, 0, 0, 0, ...
%C A130667   0, 6, 0, 0, 0, ...
%C A130667   0, 5, 1, 0, 0, ...
%C A130667   0, 0, 6, 0, 0, ...
%C A130667   0, 0, 5, 1, 0, ...
%C A130667   ...
%C A130667 The sequence divided by its aerated variant is (1, 6, 5, 0, 0, 0, ...). (End)
%H A130667 Alois P. Heinz, <a href="/A130667/b130667.txt">Table of n, a(n) for n = 1..16383</a>
%H A130667 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, pp. 27, 32-33.
%H A130667 D. E. Knuth, <a href="http://www.jstor.org/stable/27642339">Problem 11320</a>, The American Mathematical Monthly, Vol. 114, No. 9 (Nov., 2007), p. 835.
%F A130667 a(2*n) = 6*a(n) and a(2*n+1) = 5*a(n) + a(n+1).
%F A130667 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
%F A130667 a(n) = Sum_{k=0..n} 5^wt(k), where wt = A000120. - _Mike Warburton_, Mar 14 2019
%F A130667 a(n) = Sum_{k=0..floor(log_2(n))} 5^k*A360189(n-1,k). - _Alois P. Heinz_, Mar 06 2023
%p A130667 a:= proc(n) option remember;
%p A130667       `if`(n=1, 1, `if`(irem(n, 2, 'm')=0, 6*a(m), 5*a(m)+a(n-m)))
%p A130667     end:
%p A130667 seq(a(n), n=1..70); # _Alois P. Heinz_, Apr 09 2012
%t A130667 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-François Alcover_, Feb 13 2015 *)
%o A130667 (Haskell)
%o A130667 import Data.List (transpose)
%o A130667 a130667 n = a130667_list !! (n-1)
%o A130667 a130667_list = 1 : (concat $ transpose
%o A130667    [zipWith (+) vs a130667_list, zipWith (+) vs $ tail a130667_list])
%o A130667    where vs = map (* 5) a130667_list
%o A130667 -- _Reinhard Zumkeller_, Apr 18 2012
%o A130667 (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
%o A130667 (Magma) [&+[5^(2*k - Valuation(Factorial(2*k), 2)): k in [0..n]]: n in [0..50]]; // _Vincenzo Librandi_, Mar 15 2019
%Y A130667 Cf. A000120, A006046, A116520, A130665, A360189.
%K A130667 nonn
%O A130667 1,2
%A A130667 _N. J. A. Sloane_, based on a message from _Don Knuth_, Jun 23 2007