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

Author

Paul Barry, Jul 12 2005

Keywords

• easy
• nonn

Comments

Sums of antidiagonals of A083856.

Links

• 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.

Programs

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

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]