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Search: id:A058038
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| A058038 |
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Fibonacci(2*n)*Fibonacci(2*n+2). |
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+0
4
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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; listen)
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OFFSET
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0,2
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COMMENT
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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)
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REFERENCES
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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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FORMULA
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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
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)
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CROSSREFS
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Cf. A033888, A004187.
Equals A081068 - 1. Bisection of A059929, A064831 and A080097.
A058038(n)=A081068(n)-1 [From Weisenhorn Paul (paulweisenhorn(AT)online.de), May 17 2009]
Sequence in context: A067370 A094432 A104527 this_sequence A089697 A120741 A073985
Adjacent sequences: A058035 A058036 A058037 this_sequence A058039 A058040 A058041
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Jun 09 2002
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