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A363754 a(n) = Sum_{k=0..n} F(2k-1)*F(2k)*F(2k+1)/2, where F(n) is the Fibonacci number A000045(n). 1
0, 1, 16, 276, 4917, 88132, 1581196, 28372701, 509125596, 9135883240, 163936760185, 2941725767256, 52787126964456, 947226559367881, 16997290941068152, 305004010378316172, 5473074895864584141, 98210344115173624636, 1762313119177232976916, 31623425801074947486405 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This is one of the triple Fibonacci sums that were considered by Subba Rao (1953).
Taking any of the given closed-form expressions for a(n) with Fibonacci numbers, one can extend a(n) to negative indices by using the property F(-n)=(-1)^(n+1). This gives a(-n)=a(n-1).
LINKS
K. Subba Rao, Some properties of Fibonacci numbers, The American Mathematical Monthly, 60(10):680-684, December 1953.
FORMULA
a(n) = (F(2n+1)^3 + F(2n+1) - 2)/8.
a(n) = (F(6*n+3)+8*F(2*n+1)-10)/40.
a(n) = 22*a(n-1) - 77*a(n-2) + 77*a(n-3) - 22*a(n-4) + a(n-5).
G.f.: x*(1 - 6*x + x^2)/((1 - x)*(1 - 3*x + x^2)*(1 - 18*x + x^2)).
MATHEMATICA
LinearRecurrence[{22, -77, 77, -22, 1}, {0, 1, 16, 276, 4917}]
CROSSREFS
Sequence in context: A158610 A221176 A231695 * A004382 A204955 A189955
KEYWORD
nonn,easy
AUTHOR
Hans J. H. Tuenter, Jun 19 2023
STATUS
approved

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Last modified May 30 15:08 EDT 2024. Contains 372968 sequences. (Running on oeis4.)