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A058038
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Fibonacci(2n)*Fibonacci(2n+2).
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11
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0, 3, 24, 168, 1155, 7920, 54288, 372099, 2550408, 17480760, 119814915, 821223648, 5628750624, 38580030723, 264431464440, 1812440220360, 12422650078083, 85146110326224, 583600122205488, 4000054745112195, 27416783093579880, 187917426909946968
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Partial sums of A033888, i.e. a(n) = Sum_{k=0..n} Fibonacci(4*k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 09 2002
Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), 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
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 29.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (8,-8,1).
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FORMULA
| 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 (vladeta(AT)eunet.rs), Jun 09 2002
a(n) = A081068(n)-1.
Contribution from Weisenhorn Paul (paulweisenhorn(AT)online.de), May 17 2009: (Start)
a(n) is the next integer from ((3+sqrt(5))*((7+3*sqrt(5))/2)^(n-1)-6)/10
(End)
a(n)= 7*a(n-1)-a(n-2)+3, n>1 [From Gary Detlefs (gdetlefs(At)aol.com), Dec 7 2010]
a(n)= sum(Fibonacci(4k),k=0..n) [From Gary Detlefs (gdetlefs(At)aol.com), Dec 7 2010]
a(n)= (Lucas(4n+2)-3)/5, where Lucas(n)= A000032(n)[From Gary Detlefs (gdetlefs(At)aol.com), Dec 7 2010]
a(n)=(1/5)*(Fibonacci(4n+4)-Fibonacci(4n)-3) [From Gary Detlefs (gdetlefs(At)aol.com), Dec 8 2010]
a(n) = 3*A092521(n). - R. J. Mathar, Nov 03 2011
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MAPLE
| fs4:=n->sum(fibonacci(4*k), k=0..n):seq(fs4(n), n=0..21); [From Gary Detlefs (gdetlefs(At)aol.com), Dec 7 2010]
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MATHEMATICA
| Table[Fibonacci[2 n]*Fibonacci[2 n + 2], {n, 0, 100}] (* From Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
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PROG
| (MAGMA) [Fibonacci(2*n)*Fibonacci(2*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
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CROSSREFS
| Cf. A033888, A004187.
Bisection of A059929, A064831 and A080097.
Related to sum of fibonacci(kn) over n.. A000071, A099919, A027941, A138134, A053606
Sequence in context: A067370 A094432 A104527 * A089697 A120741 A073985
Adjacent sequences: A058035 A058036 A058037 * A058039 A058040 A058041
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 09 2002
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