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%I #106 Nov 14 2022 20:01:39
%S 0,1,1,-1,-3,-1,5,7,-3,-17,-11,23,45,-1,-91,-89,93,271,85,-457,-627,
%T 287,1541,967,-2115,-4049,181,8279,7917,-8641,-24475,-7193,41757,
%U 56143,-27371,-139657,-84915,194399,364229,-24569,-753027,-703889,802165,2209943,605613,-3814273
%N Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/2.
%C The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.
%C This is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i*sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30. Note that for this sequence, a(30) = 24475 = 5*5*11*89 has no primitive factors. - _T. D. Noe_, Oct 29 2003
%C Row sums of Riordan array (1/(1+2x^2), x/(1+2x^2)). - _Paul Barry_, Sep 10 2005
%C Pisano period lengths: 1, 1, 8, 2, 24, 8, 21, 2, 24, 24, 10, 8, 168, 21, 24, 4, 144, 24, 360, 24, ... - _R. J. Mathar_, Aug 10 2012
%C This is the Lucas Sequence U_n(P, Q) = U_n(1, 2). V_n(1, 2) = A002249(n). - _Raphie Frank_, Dec 25 2013
%C Note that (A002249(n)/2)^2 + 7*(a(n)/2)^2 = 2^n for all n in N. This is a specific case of the Lucas sequence identity (V_n/2)^2 - D*(U_n/2)^2 = Q^n where V_n = (a^n + b^n), U_n = (a^n - b^n)/(a - b), Q = (a*b) = 2 and D = (a - b)^2 = -7; a = (1 + sqrt(-7))/2 and b = (1 - sqrt(-7))/2. - _Raphie Frank_, Nov 26 2015
%C For the special case where |a(n)| = 1, true for n if and only if n is in {1, 2, 3, 5, 13} = {A215795(n) + 1} = {A060728(n) - 2}, then, additionally, by the Lucas sequence identity (U_2n = U_n*V_n), we have (a(2n)/2)^2 + 7*(a(n)/2)^2 = 2^n. - _Raphie Frank_, Nov 26 2015
%H Alois P. Heinz, <a href="/A107920/b107920.txt">Table of n, a(n) for n = 0..1000</a>
%H Christian Ballot, <a href="https://www.emis.de/journals/JIS/VOL21/Ballot/ballot30.html">Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions</a>, J. Int. Seq., Vol. 21 (2018), Article 18.6.5.
%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H F. Beukers, <a href="http://www.numdam.org/item?id=CM_1980__40_2_251_0">The multiplicity of binary recurrences</a>, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
%H Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, <a href="https://hal.inria.fr/inria-00072867">Existence of primitive divisors of Lucas and Lehmer numbers</a>, [Research Report] RR-3792, INRIA. 1999, pp.41, HAL Id : inria-00072867.
%H M. Mignotte, <a href="https://pmb.centre-mersenne.org/item/PMB_1989___1_A3_0/">Propriétés arithmétiques des suites récurrentes</a>, Besançon, 1988-1989, see p. 14. In French.
%H Ronald Orozco López, <a href="https://arxiv.org/abs/2211.04450">Deformed Differential Calculus on Generalized Fibonacci Polynomials</a>, arXiv:2211.04450 [math.CO], 2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas Sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-2).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F G.f.: x / (1 - x + 2*x^2).
%F a(n) = a(n-1) - 2*a(n-2).
%F a(n) = -(-1)^n*A001607(n).
%F From _Paul Barry_, Sep 10 2005: (Start)
%F a(n+1) = Sum_{k=0..n} C((n+k)/2, k)*(-2)^((n-k)/2)*(1+(-1)^(n-k))/2}.
%F a(n+1) = Sum_{k=0..floor(n/2)} C(n-k, k)(-2)^k. (End)
%F a(n+1) = Sum_{k=0..n} A109466(n,k)*2^(n-k). - _Philippe Deléham_, Oct 26 2008
%F a(n) = ((1 - i*sqrt(7))^n - (1 + i*sqrt(7))^n)*i/(2^n*sqrt(7)), where i=sqrt(-1). - _Bruno Berselli_, Jul 01 2011
%F (a(2*(A060728(n)) - 4))^2 = (A002249(A060728(n) - 2))^2 = 2^(A060728(n)) - 7 = A227078(n), the Ramanujan-Nagell squares. - _Raphie Frank_, Dec 25 2013
%F a(n) = -a(-n) * 2^n for all n in Z. - _Michael Somos_, Jan 19 2017
%F G.f.: x / (1 - x / (1 + 2*x / (1 - 2*x))). - _Michael Somos_, Jan 19 2017
%F a(n) = S(n-1, 1/sqrt(2))*(sqrt(2))^(n-1), n >= 0, with the Chebyshev S polynomials (coefficients in A049310), and S(-1, x) = 0. - _Wolfdieter Lang_, Feb 22 2018
%F a(n) = hypergeom([1-n/2, (1-n)/2], [1-n], 8) for n >= 2. - _Peter Luschny_, Feb 23 2018
%e G.f. = x + x^2 - x^3 - 3*x^4 - x^5 + 5*x^6 + 7*x^7 - 3*x^8 - 17*x^9 - 11*x^10 + ...
%p a:= n-> (Matrix([[1,1],[ -2,0]])^n)[1,2]: seq(a(n), n=0..45); # _Alois P. Heinz_, Sep 03 2008
%t LinearRecurrence[{1, -2}, {0, 1}, 50] (* _Vincenzo Librandi_, Nov 27 2015 *)
%t a[ n_] := Im[ ((1 + Sqrt[-7]) / 2)^n // FullSimplify] 2 / Sqrt[7]; (* _Michael Somos_, Jan 19 2017 *)
%t a[n_] := If[n < 2, n, Hypergeometric2F1[1 - n/2, (1 - n)/2, 1 - n, 8]];
%t Table[a[n], {n, 0, 45}] (* _Peter Luschny_, Feb 23 2018 *)
%o (PARI) {a(n) = imag(quadgen(-7)^n)};
%o (Sage) [lucas_number1(n,1,+2) for n in range(0, 46)] # _Zerinvary Lajos_, Apr 22 2009
%o (Magma) [0] cat [n le 2 select 1 else Self(n-1)-2*Self(n-2): n in [1..45]]; // _Vincenzo Librandi_, Nov 27 2015
%o (PARI) x='x+O('x^100); concat(0, Vec(x/(1-x+2*x^2))) \\ _Altug Alkan_, Dec 04 2015
%Y Cf. A001607, A002249, A049310, A060728, A077020, A167433, A169998, A215795, A227078.
%K sign,easy
%O 0,5
%A _Michael Somos_, May 28 2005
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