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 A143699 a(n) = 19*a(n-1) - 41*a(n-2) + 19*a(n-3) - a(n-4). 6
 0, 1, 19, 319, 5301, 88000, 1460701, 24245719, 402446619, 6680076601, 110880352000, 1840465787401, 30549274537419, 507077165538919, 8416803858813901, 139707705280792000, 2318961358994380101 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). A003733 = 5 * (A143699)^2. - R. K. Guy, Mar 11 2010 The sequence is the case P1 = 19, P2 = 39, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014 LINKS G. C. Greubel, Table of n, a(n) for n = 0..750 Per Hakan Lundow, Enumeration of matchings in polygraphs, Section 8.1. H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277. H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume Index to divisibility sequences Index entries for linear recurrences with constant coefficients, signature (19,-41,19,-1). FORMULA Equals sqrt(A003733(n)/5). G.f.: x*(1+x)*(1-x)/(1 - 19*x + 41*x^2 - 19*x^3 + x^4). - R. J. Mathar, Feb 09 2009 a(-n) = a(n). - Michael Somos, Feb 24 2009 a(n) = (r1^n + r2^n - r3^n - r4^n) / s1 where s1 = sqrt(205), s2 = sqrt(550 + 38*s1), s3 = 36 * sqrt(5) / s2, r1 = (19 + s1 + s2) / 4, r2 = 1/r1, r3 = (19 - s1 + s3) / 4, r4 = 1/r3. - Michael Somos, Feb 12 2012 From Peter Bala, Apr 03 2014: (Start) a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), n >= 1, where alpha = (1/4)*(19 + sqrt(205)), beta = (1/4)*(19 - sqrt(205)) and where T(n,x) denotes the Chebyshev polynomial of the first kind. a(n)= U(n-1, (sqrt(5) - 9)/4)*U(n-1, -(sqrt(5) + 9)/4) for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind. a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -39/4; 1, 19/2]. See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End) MATHEMATICA LinearRecurrence[{19, -41, 19, -1}, {0, 1, 19, 319}, 20] (* Jean-François Alcover, Dec 12 2016 *) PROG (PARI) {a(n) = n = abs(n); polcoeff( x*(1-x^2)/(1 -19*x +41*x^2 -19*x^3 +x^4) + x*O(x^n), n)} \\ Michael Somos, Feb 24 2009 (Magma) I:=[0, 1, 19, 319]; [n le 4 select I[n] else 19*Self(n-1) -41*Self(n-2) +19*Self(n-3) -Self(n-4): n in [1..30]]; // G. C. Greubel, May 31 2021 (Sage) def A143699_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( x*(1-x^2)/(1 -19*x +41*x^2 -19*x^3 +x^4) ).list() A143699_list(30) # G. C. Greubel, May 31 2021 CROSSREFS Cf. A003733, A100047. Sequence in context: A166965 A137352 A027541 * A015676 A098304 A014900 Adjacent sequences: A143696 A143697 A143698 * A143700 A143701 A143702 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, based on email from R. K. Guy, Feb 08 2009 STATUS approved

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Last modified October 4 12:52 EDT 2023. Contains 365885 sequences. (Running on oeis4.)