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Search: id:A005247
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| A005247 |
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a(n) = 3a(n-2) - a(n-4), a(0)=2, a(1)=1, a(2)=3, a(3)=2. Alternates Lucas (A000032) and Fibonacci (A000045) sequences for even and odd n.
(Formerly M0149)
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+0
5
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| 2, 1, 3, 2, 7, 5, 18, 13, 47, 34, 123, 89, 322, 233, 843, 610, 2207, 1597, 5778, 4181, 15127, 10946, 39603, 28657, 103682, 75025, 271443, 196418, 710647, 514229, 1860498, 1346269, 4870847, 3524578, 12752043, 9227465, 33385282, 24157817
(list; graph; listen)
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OFFSET
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0,1
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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.
T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.
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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(0)=2, a(1)=1; a(2)=3, a(n)=(1+a(n-1)a(n-2))/a(n-3), n >= 3. a(-n)=a(n).
G.f.: (2+x-3*x^2-x^3)/((1-x-x^2)*(1+x-x^2))
a(n)=F(n) if n odd, a(n)=L(n) if n even. a(n)=F(n+1)+(-1)^nF(n-1). - Mario Catalani (mario.catalani(AT)unito.it), Sep 20 2002
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MAPLE
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with(combinat): A005247 := n-> if n mod 2 = 1 then fibonacci(n) else fibonacci(n+1)+fibonacci(n-1); fi;
A005247:=-(z+1)*(3*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[(2 + x - 3x^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), fibonacci(n+1)+fibonacci(n-1))
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CROSSREFS
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Cf. A000032, A000045, A005013, A005013.
Adjacent sequences: A005244 A005245 A005246 this_sequence A005248 A005249 A005250
Sequence in context: A022477 A082833 A101709 this_sequence A135259 A122147 A125210
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KEYWORD
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nonn,nice,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, May 01 2000
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