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A022087
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Fibonacci sequence beginning 0 4.
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6
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0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 576, 932, 1508, 2440, 3948, 6388, 10336, 16724, 27060, 43784, 70844, 114628, 185472, 300100, 485572, 785672, 1271244, 2056916, 3328160, 5385076, 8713236, 14098312, 22811548, 36909860, 59721408, 96631268
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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. 18.
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1).
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FORMULA
| a(n) = round( (8phi-4)/5 phi^n) (works for n>2) - Thomas Baruchel, Sep 08 2004
a(n) = 4F(n) = F(n-2) + F(n) + F(n+2), with F(n) = A000045(n).
a(n) = A119457(n+2,n-1) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2006
G.f.: 4x/(1-x-x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2008]
a(n)=F(n+9)-17F(n+3), n>=0, F(n)=A000045. [From Manuel Valdivia (mvaldivia(AT)ugr.es), Dec 15 2009]
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MAPLE
| a := n-> (Matrix([[4, 0]]). Matrix([[1, 1], [1, 0]])^n)[1, 2]: seq (a(n), n=0..32); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 17 2008]
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MATHEMATICA
| a={}; b=0; c=4; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 9, 1}]; a (Vladimir Orlovsky, Jul 22 2008)
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PROG
| (PARI) a(n)=4*fibonacci(n) \\ Charles R Greathouse IV, Jun 05, 2011
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CROSSREFS
| Sequence in context: A194697 A194696 A002368 * A095294 A190100 A030168
Adjacent sequences: A022084 A022085 A022086 * A022088 A022089 A022090
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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