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A091761 a(n) = Pell(4n) / Pell(4). 3
0, 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, Jun 05 2007

A divisibility sequence, cf. R. K. Guy's post to the SeqFan list. - M. F. Hasler, Feb 05 2013

LINKS

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

R. K. Guy, A new sequence, post to the SeqFan list, Feb 05 2013

Tanya Khovanova, Recursive Sequences

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

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (34,-1).

FORMULA

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, Jun 05 2007

a(n) = 34*a(n-1)-a(n-2); a(0)=0, a(1)=1. - Philippe Deléham, Nov 03 2008

A029547(n) = a(n+1). - M. F. Hasler, Feb 05 2013

MAPLE

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

MATHEMATICA

Join[{a=0, b=1}, Table[c=34*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)

PROG

(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, Jun 05 2007

(Sage) [lucas_number1(n, 34, 1) for n in xrange(0, 16)]# Zerinvary Lajos, Nov 07 2009

CROSSREFS

A029547 is an essentially identical sequence, cf. formula.

Cf. A001109, A041085, A116108.

Sequence in context: A248163 A158696 A029547 * A264134 A264019 A009978

Adjacent sequences:  A091758 A091759 A091760 * A091762 A091763 A091764

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 06 2004

STATUS

approved

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Last modified March 24 08:12 EDT 2017. Contains 283985 sequences.