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
Links
- 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).
Programs
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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 *)
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
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