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A279100 a(n) = Sum_{k=0..n} ceiling(phi^k), where phi is the golden ratio (A001622). 0
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 (list; graph; refs; listen; history; text; internal format)
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 * A053992 A052825 A003082

Adjacent sequences:  A279097 A279098 A279099 * A279101 A279102 A279103

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Dec 06 2016

STATUS

approved

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Last modified September 21 19:57 EDT 2020. Contains 337273 sequences. (Running on oeis4.)