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A022095
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Fibonacci sequence beginning 1 5.
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16
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1, 5, 6, 11, 17, 28, 45, 73, 118, 191, 309, 500, 809, 1309, 2118, 3427, 5545, 8972, 14517, 23489, 38006, 61495, 99501, 160996, 260497, 421493, 681990, 1103483, 1785473, 2888956, 4674429, 7563385, 12237814, 19801199, 32039013, 51840212, 83879225, 135719437
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n-1)=sum(P(5;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=4. These are the sums of the SW-NE diagonals in P(5;n,k), the (5,1) Pascal triangle A093562. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of the SW-NE diagonals in the (1,4)-Pascal triangle A095666.
In general, for a Fibonacci sequence beginning with 1,b we have
a(n)=(2^(-1-n)((1-Sqrt[5])^n(1+Sqrt[5]-2b)+(1+Sqrt[5])^n(-1+Sqrt[5]+2b)))/Sqrt[5] - Herbert Kociemba(kociemba(AT)t-online.de), Dec 18 2011
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (1,1)
Tanya Khovanova, Recursive Sequences
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FORMULA
| a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=5. a(-1):=4.
G.f.: (1+4*x)/(1-x-x^2).
Row sums of triangle A131776 starting (1, 5, 6, 11, 17, 28,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007
a(n)=4*fibonacci(n)+fibonacci(n+1), n>=1 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007, corrected by R. J. Mathar, Apr 07 2011
a(n-1)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 2*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). [From Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009]
a(n)= 4*Fibonacci(n+2)-3*Fibonacci(n+1) [From Gary Detlefs (gdetlefs(AT)aol.com) Dec 21 2010]
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MAPLE
| a:=n->4*fibonacci(n)+fibonacci(n+1): seq(a(n), n=0..32); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
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MATHEMATICA
| a={}; b=1; c=5; 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)=fibonacci(n-1)+5*fibonacci(n) \\ Charles R Greathouse IV, Jun 05, 2011
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CROSSREFS
| a(n) = A101220(4, 0, n+1).
a(n) = A109754(4, n+1).
Cf. A131776.
Sequence in context: A136974 A101187 A070373 * A042531 A042839 A041373
Adjacent sequences: A022092 A022093 A022094 * A022096 A022097 A022098
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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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