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URL: https://oeis.org/A058038

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A058038
a(n) = Fibonacci(2*n)*Fibonacci(2*n+2).
15
0, 3, 24, 168, 1155, 7920, 54288, 372099, 2550408, 17480760, 119814915, 821223648, 5628750624, 38580030723, 264431464440, 1812440220360, 12422650078083, 85146110326224, 583600122205488, 4000054745112195, 27416783093579880, 187917426909946968, 1288005205276048899
OFFSET
0,2
COMMENTS
Partial sums of A033888, i.e., a(n) = Sum_{k=0..n} Fibonacci(4*k). - Vladeta Jovovic, Jun 09 2002
From Paul Weisenhorn, May 17 2009: (Start)
a(n) is the solution of the 2 equations a(n)+1=A^2 and 5*a(n)+1=B^2
which are equivalent to the Pell equation (10*a(n)+3)^2-5*(A*B)^2=4.
(End)
Numbers a(n) such as a(n)+1 and 5*a(n)+1 are perfect squares. - Sture Sjöstedt, Nov 03 2011
REFERENCES
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 29.
Ralph P. Grimaldi, Fibonacci and Catalan Numbers: An Introduction, (2012). See Exercise 22 at p. 12.
H. J. H. Tuenter, Fibonacci summation identities arising from Catalan's identity, Fib. Q., 60:4 (2022), 312-319.
LINKS
Julian West, Generating trees and the Catalan and Schröder numbers, Discrete Mathematics, Volume 146, Issues 1-3, 247-262, (1995).
FORMULA
From Vladeta Jovovic, Jun 09 2002: (Start)
a(n) = -3/5 + (1/5*sqrt(5)+3/5)*(2*1/(7+3*sqrt(5)))^n/(7+3*sqrt(5)) + (1/5*sqrt(5)-3/5)*(-2*1/(-7+3*sqrt(5)))^n/(-7+3*sqrt(5)).
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: 3*x/(1-7*x+x^2)/(1-x). (End)
a(n) = A081068(n) - 1.
a(n) is the next integer from ((3+sqrt(5))*((7+3*sqrt(5))/2)^(n-1)-6)/10. - Paul Weisenhorn, May 17 2009
From Gary Detlefs, Dec 07 2010: (Start)
a(n) = 7*a(n-1) - a(n-2) + 3, n>1.
a(n) = sum_{k=0..n} Fibonacci(4k).
a(n) = (Lucas(4n+2)-3)/5, where Lucas(n)= A000032(n). (End)
a(n) = (1/5)*(Fibonacci(4n+4) - Fibonacci(4n)-3). - Gary Detlefs, Dec 08 2010
a(n) = 3*A092521(n). - R. J. Mathar, Nov 03 2011
a(0)=0, a(1)=3, a(2)=24, a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). - Harvey P. Dale, Jul 25 2013
a(n) = A001906(n)*A001906(n+1). - R. J. Mathar, Jul 09 2019
Sum_{n>=1} 1/a(n) = 2/(3 + sqrt(5)) = A094874 - 1. - Amiram Eldar, Oct 05 2020
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jan 23 2025
From Stefano Spezia, Nov 12 2025: (Start)
a(n) = Sum_{i=1..2*n} Fibonacci(i)*Fibonacci(i+1) [K. S. Rao, 1953].
E.g.f.: exp(x)*(exp(5*x/2)*(3*cosh(3*sqrt(5)*x/2) + sqrt(5)*sinh(3*sqrt(5)*x/2)) - 3)/5. (End)
EXAMPLE
G.f. = 3*x + 24*x^2 + 168*x^3 + 1155*x^4 + 7920*x^5 + 54288*x^6 + ... - Michael Somos, Jan 23 2025
MAPLE
fs4:=n->sum(fibonacci(4*k), k=0..n):seq(fs4(n), n=0..21); # Gary Detlefs, Dec 07 2010
MATHEMATICA
Table[Fibonacci[2 n]*Fibonacci[2 n + 2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
Accumulate[Fibonacci[4*Range[0, 30]]] (* or *) LinearRecurrence[{8, -8, 1}, {0, 3, 24}, 30] (* Harvey P. Dale, Jul 25 2013 *)
PROG
(Magma) [Fibonacci(2*n)*Fibonacci(2*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
(PARI) a(n)=fibonacci(2*n)*fibonacci(2*n+2) \\ Charles R Greathouse IV, Jul 02 2013
CROSSREFS
Bisection of A059929, A064831 and A080097.
Related to sum of fibonacci(kn) over n; cf. A000071, A099919, A027941, A138134, A053606.
Sequence in context: A354677 A354678 A104527 * A089697 A120741 A361553
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 09 2002
STATUS
approved