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

Author

Paul Barry, Feb 06 2004

Keywords

• easy
• nonn

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).

Crossrefs

A029547 is an essentially identical sequence, cf. formula.

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

Programs

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

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 *)

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

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