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A022098
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Fibonacci sequence beginning 1 8.
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6
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1, 8, 9, 17, 26, 43, 69, 112, 181, 293, 474, 767, 1241, 2008, 3249, 5257, 8506, 13763, 22269, 36032, 58301, 94333, 152634, 246967, 399601, 646568, 1046169, 1692737, 2738906, 4431643, 7170549, 11602192, 18772741, 30374933, 49147674, 79522607, 128670281
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n-1)=sum(P(8;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=7. These are the SW-NE diagonals in P(8;n,k), the (8,1) Pascal triangle A093565. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs.
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LINKS
| Tanya Khovanova, Recursive Sequences
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FORMULA
| a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=8. a(-1):=7.
G.f.: (1+7*x)/(1-x-x^2).
a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 3.5*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). Offset 1. a(3)=9. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009]
a(n)= (2^(-1-n) ((1-Sqrt[5])^n (-15+Sqrt[5])+(1+Sqrt[5])^n (15+Sqrt[5])))/Sqrt[5] - Herbert Kociemba (kociemba(AT)t-online.de), Dec 18 2011
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MATHEMATICA
| a={}; b=1; c=8; 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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CROSSREFS
| a(n) = A109754(7, n+1) = A101220(7, 0, n+1).
Sequence in context: A169817 A145820 A175053 * A129659 A041130 A041307
Adjacent sequences: A022095 A022096 A022097 * A022099 A022100 A022101
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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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