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Search: id:A005013
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| A005013 |
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a(n) = 3a(n-2) - a(n-4), a(0)=0, a(1)=1, a(2)=1, a(3)=4. Alternates Fibonacci (A000045) and Lucas (A000032) sequences for even and odd n.
(Formerly M3214)
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+0
5
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| 0, 1, 1, 4, 3, 11, 8, 29, 21, 76, 55, 199, 144, 521, 377, 1364, 987, 3571, 2584, 9349, 6765, 24476, 17711, 64079, 46368, 167761, 121393, 439204, 317811, 1149851, 832040, 3010349, 2178309, 7881196, 5702887, 20633239, 14930352, 54018521
(list; graph; listen)
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OFFSET
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0,4
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REFERENCES
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S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
A. F. Horadam, R. P. Loh and A. G. Shannon, Divisibility properties of some Fibonacci-type sequences, pp. 55-64 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..500
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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a(1)=a(2)=1, a(3)=4, a(n)=(a(n-1)a(n-2)-1)/a(n-3), n >= 4. a(-n)=-a(n).
a(n)=F(n) if n even, a(n)=L(n) if n odd. a(n)=F(n+1)+(-1)^(n+1)F(n-1). - Mario Catalani (mario.catalani(AT)unito.it), Sep 20 2002
G.f.: (x+x^2+x^3)/((1-x-x^2)*(1+x-x^2))
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MAPLE
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with(combinat): A005013 := n-> if n mod 2 = 0 then fibonacci(n) else fibonacci(n+1)+fibonacci(n-1); fi;
A005013:=z*(z**2+z+1)/(z**2+z-1)/(z**2-z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
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CoefficientList[Series[(x + x^2 + x^3)/(1 - 3x^2 + x^4), {x, 0, 40}], x]
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PROGRAM
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(PARI) a(n)=if(n%2, fibonacci(n+1)+fibonacci(n-1), fibonacci(n))
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CROSSREFS
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Cf. A000032, A000045, A005247.
Adjacent sequences: A005010 A005011 A005012 this_sequence A005014 A005015 A005016
Sequence in context: A065763 A100492 A072183 this_sequence A086564 A080777 A001166
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KEYWORD
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nonn,easy
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AUTHOR
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njas
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EXTENSIONS
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Additional comments from Michael Somos, Jun 01 2000.
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