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 A091761 a(n) = Pell(4n) / Pell(4). 4
 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 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 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(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 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. A001109, A041085, A056771, 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 May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)