A091761 a(n) = Pell(4n) / Pell(4).

0, 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775, 92665403695822344828176, 3147895910861898495432209

Offset

0,3

Comments

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

a(n) gives all nonnegative solutions of the Pell equation b(n)^2 - 32*(3*a(n))^2 = +1, together with b(n) = A056771(n). - Wolfdieter Lang, Mar 09 2019

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/(1-34*x+x^2).

a(n) = A000129(4n)/A000129(4).

a(n) = ((1+sqrt(2))^(4n) - (1-sqrt(2))^(4n))*sqrt(2)/48.

From M. F. Hasler, Jun 05 2007: (Start)

a(n) = n (mod 2^m) for any m >= 0.

a(n) = sinh(4*n*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). (End)

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

a(n) = sqrt((A056771(n)^2 - 1)/(32*9)), n >= 0. See the Pell remark above. - Wolfdieter Lang, Mar 11 2019

E.g.f.: exp(17*x)*sinh(12*sqrt(2)*x)/(12*sqrt(2)). - Stefano Spezia, Apr 16 2023

a(n) = A002965(8*n)/12. - Gerry Martens, Jul 14 2023

Maple

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

Mathematica

LinearRecurrence[{34, -1}, {0, 1}, 20] (* G. C. Greubel, Mar 11 2019 *)

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 range(0, 16)]# Zerinvary Lajos, Nov 07 2009
(Magma) I:=[0, 1]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // G. C. Greubel, Mar 11 2019

Crossrefs

A029547 is an essentially identical sequence, cf. formula.

Cf. A000129, A001109, A002965, A029547, A041085, A056771, A116108.

Sequence in context: A248163 A158696 A029547 * A264134 A264019 A009978

Adjacent sequences: A091758 A091759 A091760 * A091762 A091763 A091764

Keyword

easy,nonn,changed

Author

Paul Barry, Feb 06 2004

Status

approved