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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).
1
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
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
Sequence in context: A123209 A123604 A321306 * A047756 A047767 A303588
KEYWORD
nonn
EXTENSIONS
More terms from James A. Sellers, May 03 2000
STATUS
approved