A020957 a(n) = Sum_{k >= 1} floor(2*tau^(n-k)).

3, 6, 11, 19, 32, 54, 89, 147, 240, 392, 637, 1035, 1678, 2720, 4405, 7133, 11546, 18688, 30243, 48941, 79194, 128146, 207351, 335509, 542872, 878394, 1421279, 2299687, 3720980, 6020682, 9741677, 15762375, 25504068, 41266460, 66770545

Offset

1,1

Links

Table of n, a(n) for n=1..35.

C. Kimberling, Problem 10520, Amer. Math. Mon. 103 (1996) p. 347.

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

Formula

a(n) = (1/4)*(8*Lucas(n+1) - 2n - 5 + (-1)^n), n > 1.

G.f.: x*(x^5 + x^4 - 4*x^2 + 3)/((1 - x)*(1 - x^2)*(1 - x - x^2)).

E.g.f.: (4*exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)) - (2 + x)*cosh(x) - (3 + x)*sinh(x) - 2*(1 + x))/2. - Stefano Spezia, Feb 24 2023

Mathematica

CoefficientList[Series[x (x^5+x^4-4x^2+3)/((1-x)(1-x^2)(1-x-x^2)), {x, 0, 30}], x] (* Harvey P. Dale, May 10 2018 *)

Crossrefs

Cf. A001622 (tau), A020958.

Sequence in context: A326957 A116557 A001911 * A179006 A281573 A262987

Adjacent sequences: A020954 A020955 A020956 * A020958 A020959 A020960

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Harvey P. Dale, May 10 2018

a(29)-a(32) corrected and more terms from Sean A. Irvine, May 05 2019

Name edited by Michel Marcus, Jul 06 2019

Status

approved