A110113 Diagonal sums of A083856.

0, 1, 2, 3, 5, 9, 17, 34, 71, 154, 346, 802, 1914, 4693, 11800, 30379, 79963, 214925, 589223, 1645994, 4681037, 13541446, 39817560, 118925810, 360577616, 1109158545, 3459636358, 10936941299, 35026082521, 113588037953

Offset

0,3

Comments

Sums of antidiagonals of A083856.

Links

Table of n, a(n) for n=0..29.

A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, 21(2) (2015), 35-42.

Formula

a(n) = Sum_{k = 0..n} ((1 + sqrt(4*(n - k) + 1))/2)^k / sqrt(4*(n - k) + 1) - ((1 -sqrt(4*(n - k) + 1))/2)^k / sqrt(4*(n - k) + 1). [Corrected by Petros Hadjicostas, Dec 26 2019]

Maple

T := proc(n, k) local v; option remember; if 0 <= n and k = 0 then v := 0; end if; if 0 <= n and k = 1 then v := 1; end if; if 0 <= n and 2 <= k then v := T(n, k - 1) + n*T(n, k - 2); end if; v; end proc;
a := proc(n) local k; add(T(n - k, k), k = 0 .. n); end proc;
seq(a(n), n = 0..40); # Petros Hadjicostas, Dec 26 2019

Crossrefs

Sequence in context: A061902 A166286 A179807 * A341960 A137155 A014227

Adjacent sequences: A110110 A110111 A110112 * A110114 A110115 A110116

Keyword

easy,nonn

Author

Paul Barry, Jul 12 2005

Status

approved