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a(n) = Sum_{k=0..n} ceiling(phi^k), where phi is the golden ratio (A001622).
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%I #5 Feb 16 2025 08:33:37

%S 1,3,6,11,18,30,48,78,125,202,325,525,847,1369,2212,3577,5784,9356,

%T 15134,24484,39611,64088,103691,167771,271453,439215,710658,1149863,

%U 1860510,3010362,4870860,7881210,12752057,20633254,33385297,54018537,87403819,141422341,228826144,370248469,599074596

%N a(n) = Sum_{k=0..n} ceiling(phi^k), where phi is the golden ratio (A001622).

%C Partial sums of A169986.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,0,1).

%F G.f.: (1 + x - x^2 - x^3 - x^4)/((1 - x)^2*(1 - 2*x^2 - x^3)).

%F a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5).

%F a(n) = (10*n - 5*(-1)^n + 2^(1-n)*sqrt(5)*(5 + 3*sqrt(5))*(1 + sqrt(5))^n + sqrt(5)*2^(1-n)*(3*sqrt(5) - 5) *(1 - sqrt(5))^n - 35)/20.

%F a(n) ~ phi^(n+2).

%t Accumulate[Table[Ceiling[GoldenRatio^n], {n, 0, 40}]]

%t LinearRecurrence[{2, 1, -3, 0, 1}, {1, 3, 6, 11, 18}, 41]

%Y Cf. A001622, A020956, A169986.

%K nonn,easy,changed

%O 0,2

%A _Ilya Gutkovskiy_, Dec 06 2016