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%I #18 Dec 26 2019 04:46:22
%S 0,1,2,3,5,9,17,34,71,154,346,802,1914,4693,11800,30379,79963,214925,
%T 589223,1645994,4681037,13541446,39817560,118925810,360577616,
%U 1109158545,3459636358,10936941299,35026082521,113588037953
%N Diagonal sums of A083856.
%C Sums of antidiagonals of A083856.
%H A. G. Shannon and J. V. Leyendekkers, <a href="http://nntdm.net/volume-21-2015/number-2/35-42/">The Golden Ratio family and the Binet equation</a>, Notes on Number Theory and Discrete Mathematics, 21(2) (2015), 35-42.
%F 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]
%p 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;
%p a := proc(n) local k; add(T(n - k, k), k = 0 .. n); end proc;
%p seq(a(n), n = 0..40); # _Petros Hadjicostas_, Dec 26 2019
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jul 12 2005
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