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A091761 a(n) = Pell(4n) / Pell(4). 5
0, 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775, 92665403695822344828176, 3147895910861898495432209 (list; graph; refs; listen; history; text; internal format)
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
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).
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(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). (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.
Sequence in context: A248163 A158696 A029547 * A264134 A264019 A009978
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 06 2004
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)