The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A107920 Lucas and Lehmer numbers with parameters (1+-sqrt(-7))/2. 31
 0, 1, 1, -1, -3, -1, 5, 7, -3, -17, -11, 23, 45, -1, -91, -89, 93, 271, 85, -457, -627, 287, 1541, 967, -2115, -4049, 181, 8279, 7917, -8641, -24475, -7193, 41757, 56143, -27371, -139657, -84915, 194399, 364229, -24569, -753027, -703889, 802165, 2209943, 605613, -3814273 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs. 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 Row sums of Riordan array (1/(1+2x^2),x/(1+2x^2)). - Paul Barry, Sep 10 2005 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 This is the Lucas Sequence U_n(P, Q) = U_n(1, 2). V_n(1, 2) = A002249(n). - Raphie Frank, Dec 25 2013 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 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Christian Ballot, Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions, J. Int. Seq., Vol. 21 (2018), Article 18.6.5. Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020. F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259. Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, Existence of primitive divisors of Lucas and Lehmer numbers, [Research Report] RR-3792, INRIA. 1999, pp.41, HAL Id : inria-00072867. M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French. Eric Weisstein's World of Mathematics, Lehmer Number Wikipedia, Lucas Sequence Index entries for linear recurrences with constant coefficients, signature (1,-2). FORMULA G.f.: x / (1 - x + 2*x^2). a(n) = a(n-1) - 2*a(n-2). a(n) = -(-1)^n*A001607(n). From Paul Barry, Sep 10 2005: (Start) a(n+1) = Sum_{k=0..n} C((n+k)/2, k)*(-2)^((n-k)/2)*(1+(-1)^(n-k))/2}. a(n+1) = Sum_{k=0..floor(n/2)} C(n-k, k)(-2)^k. (End) a(n+1) = Sum_{k=0..n} A109466(n,k)*2^(n-k). - Philippe Deléham, Oct 26 2008 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 (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 a(n) = -a(-n) * 2^n for all n in Z. - Michael Somos, Jan 19 2017 G.f.: x / (1 - x / (1 + 2*x / (1 - 2*x))). - Michael Somos, Jan 19 2017 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 a(n) = hypergeom([1-n/2, (1-n)/2], [1-n], 8) for n >= 2. - Peter Luschny, Feb 23 2018 EXAMPLE 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 + ... MAPLE a:= n-> (Matrix([[1, 1], [ -2, 0]])^n)[1, 2]: seq(a(n), n=0..45); # Alois P. Heinz, Sep 03 2008 MATHEMATICA LinearRecurrence[{1, -2}, {0, 1}, 50] (* Vincenzo Librandi, Nov 27 2015 *) a[ n_] := Im[ ((1 + Sqrt[-7]) / 2)^n // FullSimplify] 2 / Sqrt; (* Michael Somos, Jan 19 2017 *) a[n_] := If[n < 2, n, Hypergeometric2F1[1 - n/2, (1 - n)/2, 1 - n, 8]]; Table[a[n], {n, 0, 45}] (* Peter Luschny, Feb 23 2018 *) PROG (PARI) {a(n) = imag(quadgen(-7)^n)}; (Sage) [lucas_number1(n, 1, +2) for n in range(0, 46)] # Zerinvary Lajos, Apr 22 2009 (MAGMA)  cat [n le 2 select 1 else Self(n-1)-2*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 27 2015 (PARI) x='x+O('x^100); concat(0, Vec(x/(1-x+2*x^2))) \\ Altug Alkan, Dec 04 2015 CROSSREFS Cf. A001607, A002249, A049310, A060728, A077020, A167433, A169998, A215795, A227078. Sequence in context: A001607 A167433 A077020 * A169998 A326729 A171998 Adjacent sequences:  A107917 A107918 A107919 * A107921 A107922 A107923 KEYWORD sign,easy AUTHOR Michael Somos, May 28 2005 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 26 13:40 EDT 2021. Contains 346294 sequences. (Running on oeis4.)