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Search: id:A098149
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| A098149 |
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a(0)=-1, a(1)=-1, a(n)=-3*a(n-1)-a(n-2) for n>1. |
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+0
4
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| -1, -1, 4, -11, 29, -76, 199, -521, 1364, -3571, 9349, -24476, 64079, -167761, 439204, -1149851, 3010349, -7881196, 20633239, -54018521, 141422324, -370248451, 969323029, -2537720636, 6643838879, -17393796001, 45537549124, -119218851371
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Sequence relates bisections of Lucas and Fibonacci numbers.
2*a(n) + A098150(n) = 8*(-1)^(n+1)*A001519(n) - (-1)^(n+1)*A005248(n+1). Apparently, if (z(n)) is any sequence of integers (not all zero) satisfying the formula z(n) = 2(z(n-2) - z(n-1)) + z(n-3) then |z(n+1)/z(n)| -> golden ratio phi + 1 = (3+sqrt(5))/2
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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FORMULA
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G.f.: -(1+4*x)/(1+3*x+x^2).
a(n) = 2(a(n-2) - a(n-1)) + a(n-3), with a(0) = a(1) = -1 and a(2) = 4.
-a(n+1)= Sum_{k, 0<=k<=n}(-5)^k*Binomial(n+k,n-k)= Sum_{k, 0<=k<=n}(-5)^k*A085478(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2006
a(n)=-(1/2)*[(-3/2)-(1/2)*sqrt(5)]^n+(1/2)*[(-3/2)-(1/2)*sqrt(5)]^n*sqrt(5)-(1/2)*[(-3/2)+(1/2) *sqrt(5)]^n*sqrt(5)-(1/2)*[(-3/2)+(1/2)*sqrt(5)]^n, with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 19 2008]
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MATHEMATICA
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a[0] = a[1] = -1; a[2] = 4; a[n_] := a[n] = 2(a[n - 2] - a[n - 1]) + a[n - 3]; Table[ a[n], {n, 0, 27}] (from Robert G. Wilson v Sep 01 2004)
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CROSSREFS
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Cf. A098150, A001519, A005248.
Adjacent sequences: A098146 A098147 A098148 this_sequence A098150 A098151 A098152
Sequence in context: A027970 A027972 A002878 this_sequence A124861 A110579 A084378
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KEYWORD
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easy,sign
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AUTHOR
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Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 29 2004
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Sep 1 2004
Simpler definition and generating function from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2006
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