The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60, we have over 367,000 sequences, and we’ve crossed 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A107920 Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/2. 30
 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. Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022. Eric Weisstein's World of Mathematics, Lehmer Number Wikipedia, Lucas Sequence Index entries for linear recurrences with constant coefficients, signature (1,-2). Index entries for sequences related to Chebyshev polynomials. 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[7]; (* 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) [0] 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 1 21:21 EST 2023. Contains 367502 sequences. (Running on oeis4.)