0,3

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

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. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.

Index entries for sequences related to linear recurrences with constant coefficients

Joerg Arndt, Fxtbook

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. Bremner and N. Tzanakis, Lucas sequences whose 8th term is a square

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 437

a(n)={[ (1+sqrt(17))/2 ]^(n+1) - [ (1-sqrt(17))/2 ]^(n+1)}/sqrt(17).

a(n+1)=sum(k=0, ceil(n/2), 4^k*binomial(n-k, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 06 2004

G.f.: 1/(1-x-4x^2).

a(n)=sum{k=0..n, binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))2^(n-k)/2}; - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005

A102446/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008

a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*(-4)^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2008]

a(n)=Product[(1 + m*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2008]

Limiting ratio (m=16)=(1 + Sqrt[1 + m])/2=2.5615528128088303. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2008]

A006131:=-1/(-1+z+4*z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

m = 16; f[n_] = Product[(1 + m*Cos[k*Pi/n]^2), {k, 1, Floor[(n - 1)/2]}]; Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 15}]; N[%] [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 21 2008]

sage: from sage.combinat.sloane_functions import recur_gen2b sage: it = recur_gen2b(2, 2, 1, 4, lambda n: 0) sage: [it.next()/2 for i in range(29)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008

(Other) sage: [lucas_number1(n, 1, -4) for n in xrange(1, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

Cf. A006130, A015440.

Cf. A026581, A026583, A026597, A026599, A052923.

Cf. A102446.

Sequence in context: A147367 A147230 A163779 this_sequence A049602 A119031 A034435

Adjacent sequences: A006128 A006129 A006130 this_sequence A006132 A006133 A006134

nonn

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

More terms from Roger Bagula, Sep 26 2006