A279100 a(n) = Sum_{k=0..n} ceiling(phi^k), where phi is the golden ratio (A001622).

1, 3, 6, 11, 18, 30, 48, 78, 125, 202, 325, 525, 847, 1369, 2212, 3577, 5784, 9356, 15134, 24484, 39611, 64088, 103691, 167771, 271453, 439215, 710658, 1149863, 1860510, 3010362, 4870860, 7881210, 12752057, 20633254, 33385297, 54018537, 87403819, 141422341, 228826144, 370248469, 599074596

Offset: 0

Author

Ilya Gutkovskiy, Dec 06 2016

Keywords

• nonn
• easy

Comments

Partial sums of A169986.

Links

• Eric Weisstein's World of Mathematics, Golden Ratio
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Crossrefs

Cf. A001622, A020956, A169986.

Programs

Mathematica

Accumulate[Table[Ceiling[GoldenRatio^n], {n, 0, 40}]]
LinearRecurrence[{2, 1, -3, 0, 1}, {1, 3, 6, 11, 18}, 41]

Formula

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

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

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.

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