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Search: id:A022103
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| A022103 |
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Fibonacci sequence beginning 1 13. |
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+0
3
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| 1, 13, 14, 27, 41, 68, 109, 177, 286, 463, 749, 1212, 1961, 3173, 5134, 8307, 13441, 21748, 35189, 56937, 92126, 149063, 241189, 390252, 631441, 1021693, 1653134, 2674827, 4327961, 7002788, 11330749
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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(13;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=12. These are the SW-NE diagonals in P(13;n,k), the (13,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 29 2004. Proof via recursion relations and comparison of inputs.
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LINKS
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Tanya Khovanova, Recursive Sequences
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FORMULA
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a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=13. a(-1):=12.
G.f.: (1+12*x)/(1-x-x^2).
a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 6*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). Offset 1. a(3)=14. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009]
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MATHEMATICA
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a={}; b=1; c=13; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 12, 1}]; a (Vladimir Orlovsky, Jul 23 2008)
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CROSSREFS
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a(n) = A109754(12, n+1) = A101220(12, 0, n+1).
Sequence in context: A022803 A112653 A015905 this_sequence A041342 A041344 A041340
Adjacent sequences: A022100 A022101 A022102 this_sequence A022104 A022105 A022106
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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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