0,3

A000129(kn)/A000129(k)=((sqrt(2)-1)^k(-1)^k-(sqrt(2)+1)^k)((sqrt(2)-1)^(kn)(-1)^(kn)-(sqrt(2)+1)^(kn))/((sqrt(2)-1)^(2k)+(sqrt(2)+1)^(2k)-2(-1)^k)

All squares of the form (3m-1)^3 + (3m)^3 + (3m+1)^3 (cf. A116108) are given for m = 24 b, where b is a square of this sequence. From Ribenboim & McDaniel, it follows there are no squares > 1 in this sequence. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jun 05 2007

Paulo Ribenboim and Wayne L. McDaniel: "The Square Terms in Lucas Sequences", Journal of Number Theory 58, 104-123 (1996).

M. F. Hasler, Table of n, a(n) for n = 0..99

Tanya Khovanova, Recursive Sequences

G.f.: x/(1-34x+x^2); a(n)=A000129(4n)/A000129(4); a(n)=((1+sqrt(2))^(4n)-(1-sqrt(2))^(4n))sqrt(2)/48.

a(n) = n (mod 2^m) for any m>=0. a(n) = sinh(4n*log(sqrt(2)+1)/(12 sqrt(2)) a(n) = A[1,1], first element of the 2 X 2 matrix A = (34,1;-1,0)^(n-1) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jun 05 2007

a(n)=34*a(n-1)-a(n-2); a(0)=0, a(1)=1. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]

with (combinat):seq(fibonacci(4*n, 2)/12, n=0..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2008

(PARI) A091761(n, x=[ -1, 17], A=[17, 72*4; 1, 17]) = vector(n, i, (x*=A)[1]) - M. F. Hasler, May 26 2007

(PARI) A091761(n)=([34, 1; -1, 0]^(n-1))[1, 1] - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Jun 05 2007

(Other) sage: [lucas_number1(n, 34, 1) for n in xrange(0, 16)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]

A029547 is an essentially identical sequence.

Cf. A001109, A041085, A116108.

Sequence in context: A075292 A158696 A029547 this_sequence A009978 A041545 A167258

Adjacent sequences: A091758 A091759 A091760 this_sequence A091762 A091763 A091764

easy,nonn,new

Paul Barry (pbarry(AT)wit.ie), Feb 06 2004