A058038 - OEIS

%I #52 Dec 29 2022 15:45:47

%S 0,3,24,168,1155,7920,54288,372099,2550408,17480760,119814915,

%T 821223648,5628750624,38580030723,264431464440,1812440220360,

%U 12422650078083,85146110326224,583600122205488,4000054745112195,27416783093579880,187917426909946968

%N a(n) = Fibonacci(2*n)*Fibonacci(2*n+2).

%C Partial sums of A033888, i.e., a(n) = Sum_{k=0..n} Fibonacci(4*k). - _Vladeta Jovovic_, Jun 09 2002

%C From _Paul Weisenhorn_, May 17 2009: (Start)

%C a(n) is the solution of the 2 equations a(n)+1=A^2 and 5*a(n)+1=B^2

%C which are equivalent to the Pell equation (10*a(n)+3)^2-5*(A*B)^2=4.

%C (End)

%C Numbers a(n) such as a(n)+1 and 5*a(n)+1 are perfect squares. - _Sture Sjöstedt_, Nov 03 2011

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 29.

%D H. J. H. Tuenter, Fibonacci summation identities arising from Catalan's identity, Fib. Q., 60:4 (2022), 312-319.

%H Harvey P. Dale, <a href="/A058038/b058038.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).

%F 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)). Recurrence: a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). G.f.: 3*x/(1-7*x+x^2)/(1-x). - _Vladeta Jovovic_, Jun 09 2002

%F a(n) = A081068(n) - 1.

%F a(n) is the next integer from ((3+sqrt(5))*((7+3*sqrt(5))/2)^(n-1)-6)/10. - _Paul Weisenhorn_, May 17 2009

%F a(n) = 7*a(n-1) - a(n-2) + 3, n>1. - _Gary Detlefs_, Dec 07 2010

%F a(n) = sum_{k=0..n} Fibonacci(4k). - _Gary Detlefs_, Dec 07 2010

%F a(n) = (Lucas(4n+2)-3)/5, where Lucas(n)= A000032(n). - _Gary Detlefs_, Dec 07 2010

%F a(n) = (1/5)*(Fibonacci(4n+4) - Fibonacci(4n)-3). - _Gary Detlefs_, Dec 08 2010

%F a(n) = 3*A092521(n). - _R. J. Mathar_, Nov 03 2011

%F 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

%F a(n) = A001906(n)*A001906(n+1). - _R. J. Mathar_, Jul 09 2019

%F Sum_{n>=1} 1/a(n) = 2/(3 + sqrt(5)) = A094874 - 1. - _Amiram Eldar_, Oct 05 2020

%p fs4:=n->sum(fibonacci(4*k),k=0..n):seq(fs4(n),n=0..21); # _Gary Detlefs_, Dec 07 2010

%t Table[Fibonacci[2 n]*Fibonacci[2 n + 2], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 01 2011 *)

%t Accumulate[Fibonacci[4*Range[0,30]]] (* or *) LinearRecurrence[{8,-8,1},{0,3,24},30] (* _Harvey P. Dale_, Jul 25 2013 *)

%o (Magma) [Fibonacci(2*n)*Fibonacci(2*n+2): n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011

%o (PARI) a(n)=fibonacci(2*n)*fibonacci(2*n+2) \\ _Charles R Greathouse IV_, Jul 02 2013

%Y Cf. A033888, A004187, A094874.

%Y Bisection of A059929, A064831 and A080097.

%Y Related to sum of fibonacci(kn) over n; cf. A000071, A099919, A027941, A138134, A053606.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jun 09 2002