A024309 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = (natural numbers >= 2), t = (Fibonacci numbers).

2, 2, 7, 12, 27, 43, 85, 138, 253, 409, 718, 1162, 1984, 3210, 5383, 8710, 14433, 23353, 38391, 62118, 101577, 164355, 267804, 433316, 704382, 1139714, 1849751, 2992960, 4852471, 7851463, 12720713, 20582546, 33332005, 53932317, 87313418

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

G.f.: (2 -x^2 +3*x^3 -2*x^5 -x^7)/((1-x-x^2)*(1-x^2-x^4)^2). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

From G. C. Greubel, Feb 21 2022: (Start)

a(n) = Sum_{j=2..floor((n+3)/2)} j*Fibonacci(n-j+2).

a(n) = Lucas(n+3) - (m+1)*Fibonacci(n-m+5) + m*Fibonacci(n-m+4), where m = floor((n+3)/2).

a(2*n) = Lucas(2*n+3) - Lucas(n+3) - n*Fibonacci(n+2), n >= 1.

a(2*n+1) = Lucas(2*n+4) - Lucas(n+3) - (n+1)*Fibonacci(n+2), n >= 0. (End)

Mathematica

With[{m=Floor[(n+3)/2]}, Table[LucasL[n+3] -(m+1)*Fibonacci[n-m+5] +m*Fibonacci[n -m+4], {n, 50}]] (* G. C. Greubel, Feb 21 2022 *)

Prog

(PARI) a(n) = sum(j=1, (n+1)\2, (j+1)*fibonacci(n+1-j)); \\ Michel Marcus, Sep 30 2013
(Sage) [sum(j*fibonacci(n+2-j) for j in (2..floor((n+3)/2))) for n in (1..50)] # G. C. Greubel, Feb 21 2022
(Magma) [(&+[j*Fibonacci(n+2-j): j in [2..Floor((n+3)/2)]]) : n in [1..50]]; // G. C. Greubel, Feb 21 2022

Crossrefs

Cf. A000032, A000045.

Sequence in context: A123209 A123604 A321306 * A047756 A047767 A303588

Adjacent sequences: A024306 A024307 A024308 * A024310 A024311 A024312

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from James A. Sellers, May 03 2000

Status

approved