0,1

A. F. Horadam, Generating identities for generalized Fibonacci and Lucas triples, Fib. Quart., 15 (1977), 289-292.

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

Tanya Khovanova, Recursive Sequences

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.

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

a(n) = [(3 + sqrt13)/2]^n + [(3 - sqrt13)/2]^n; A006190(n-2) + A006190(n) = a(n-1); [a(n)]^2 - 13[A006190(n)]^2 = 4(-1)^n. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2003

E.g.f. : 2exp(3x/2)cosh(sqrt(13)x/2); a(n)=2^(1-n)sum{k=0..floor(n/2), C(n, 2k)13^k3^(n-2k)}. a(n)=2T(n, 3i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jan 02 2009 (Start): fract(((3+sqrt(13))/2)^n))=(1/2)*(1+(-1)^n)-(-1)^n*((3+sqrt(13))/2)^(-n)=(1/2)*(1+(-1)^n)-((3-sqrt(13))/2)^n.

See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which suffice x-x^(-1)=floor(x).

a(n) = nint(((3+sqrt(13))/2)^n) for n>0. (End)

A006497:=(-2+3*z)/(-1+3*z+z**2); [S. Plouffe in his 1992 dissertation.]

Table[LucasL[n, 3], {n, 0, 26}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]

(Other) sage: [lucas_number2(n, 3, -1) for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]

Cf. A006190.

Cf. A100230.

Cf. A001622, A014176, A080039, A098316.

Sequence in context: A144056 A062630 A159458 this_sequence A038912 A019361 A093804

Adjacent sequences: A006494 A006495 A006496 this_sequence A006498 A006499 A006500

nonn,easy

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

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 14 2004