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 A041219 Denominators of continued fraction convergents to sqrt(120). 3
 1, 1, 21, 22, 461, 483, 10121, 10604, 222201, 232805, 4878301, 5111106, 107100421, 112211527, 2351330961, 2463542488, 51622180721, 54085723209, 1133336644901, 1187422368110, 24881784007101, 26069206375211, 546265911511321, 572335117886532, 11992968269241961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 20 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Eric W. Weisstein, MathWorld: Lehmer Number Index entries for linear recurrences with constant coefficients, signature (0,22,0,-1). FORMULA From Colin Barker, Jul 15 2012: (Start) a(n) = 22*a(n-2) - a(n-4). G.f.: (1+x-x^2)/(1-22*x^2+x^4). (End) From Peter Bala, May 28 2014: (Start) The following remarks assume an offset of 1. Let alpha = sqrt(5) + sqrt(6) and beta = sqrt(5) - sqrt(6) be the roots of the equation x^2 - sqrt(20)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even. a(n) = Product_{k = 1..floor((n-1)/2)} ( 20 + 4*cos^2(k*Pi/n) ). Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 20*a(2*n) + a(2*n - 1). (End) MATHEMATICA Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[120], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *) Denominator[Convergents[Sqrt[120], 30]] (* Harvey P. Dale, Mar 14 2013 *) CoefficientList[Series[(1 + x - x^2)/(1 - 22 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 24 2013 *) CROSSREFS Cf. A041218, A002530. Sequence in context: A041916 A041918 A068325 * A041920 A041921 A041922 Adjacent sequences: A041216 A041217 A041218 * A041220 A041221 A041222 KEYWORD nonn,frac,easy,less AUTHOR N. J. A. Sloane STATUS approved

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Last modified September 19 12:50 EDT 2024. Contains 376012 sequences. (Running on oeis4.)