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Search: id:A022096
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| A022096 |
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Fibonacci sequence beginning 1 6. |
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+0
8
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| 1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10569, 17101, 27670, 44771, 72441, 117212, 189653, 306865, 496518, 803383, 1299901, 2103284, 3403185, 5506469, 8909654
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n-1)=sum(P(6;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=5. These are the sums of the SW-NE diagonals in P(6;n,k), the (6,1) Pascal triangle A093563. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of SW-NE diagonals in (1,5)-Pacal triangle A096940.
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LINKS
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Tanya Khovanova, Recursive Sequences
Dan Sewell Ward, Modified Fibonacci Sequence.
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FORMULA
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a(n) = a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=6. a(-1):=5.
G.f.: (1+5*x)/(1-x-x^2).
Row sums of triangle A131777: (1, 6, 7, 13, 20,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 14 2007
a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 2.5*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). Offset 1. a(3)=7. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009]
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MATHEMATICA
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a={}; b=1; c=6; 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
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a(n) = A101220(5, 0, n+1).
a(n) = A109754(5, n+1).
Cf. A131777.
Sequence in context: A127020 A154662 A070398 this_sequence A041175 A041074 A041749
Adjacent sequences: A022093 A022094 A022095 this_sequence A022097 A022098 A022099
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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).
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