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

0,2

Comments

Partial sums of A169986.

Links

Table of n, a(n) for n=0..40.

Eric Weisstein's World of Mathematics, Golden Ratio

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

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).

Mathematica

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

Crossrefs

Cf. A001622, A020956, A169986.

Sequence in context: A152074 A123629 A212484 * A347415 A053992 A052825

Adjacent sequences: A279097 A279098 A279099 * A279101 A279102 A279103

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Dec 06 2016

Status

approved