0,4

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

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.

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))

a(n)=Product[(1 + 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]. [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)tahoo.com), Nov 26 2008]

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)); [S. Plouffe in his 1992 dissertation.]

CoefficientList[Series[(x + x^2 + x^3)/(1 - 3x^2 + x^4), {x, 0, 40}], x]

f[n_] = Product[(1 + 4*Sin[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; a = Table[f[n], {n, 0, 30}]; Round[a]; FullSimplify[ExpandAll[a]] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)tahoo.com), Nov 26 2008]

(PARI) a(n)=if(n%2, fibonacci(n+1)+fibonacci(n-1), fibonacci(n))

Cf. A000032, A000045, A005247.

Sequence in context: A065763 A100492 A072183 this_sequence A086564 A080777 A001166

Adjacent sequences: A005010 A005011 A005012 this_sequence A005014 A005015 A005016

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Somos, Jun 01 2000.