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A091761
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a(n) = Pell(4n) / Pell(4).
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5
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0, 1, 34, 1155, 39236, 1332869, 45278310, 1538129671, 52251130504, 1775000307465, 60297759323306, 2048348816684939, 69583562007964620, 2363792759454112141, 80299370259431848174, 2727814796061228725775, 92665403695822344828176, 3147895910861898495432209
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: x/(1-34*x+x^2).
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). (End)
E.g.f.: exp(17*x)*sinh(12*sqrt(2)*x)/(12*sqrt(2)). - Stefano Spezia, Apr 16 2023
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MAPLE
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with (combinat):seq(fibonacci(4*n, 2)/12, n=0..17); # Zerinvary Lajos, Apr 21 2008
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MATHEMATICA
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LinearRecurrence[{34, -1}, {0, 1}, 20] (* G. C. Greubel, Mar 11 2019 *)
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PROG
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(PARI) A091761(n, x=[ -1, 17], A=[17, 72*4; 1, 17]) = vector(n, i, (x*=A)[1]) \\ M. F. Hasler, May 26 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
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CROSSREFS
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A029547 is an essentially identical sequence, cf. formula.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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