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 A078366 A Chebyshev S-sequence with Diophantine property. 7
 1, 17, 288, 4879, 82655, 1400256, 23721697, 401868593, 6808044384, 115334885935, 1953885016511, 33100710394752, 560758191694273, 9499788548407889, 160935647131239840, 2726406212682669391 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) gives the general (positive integer) solution of the Pell equation b^2 - 285*a^2 = +4 with companion sequence b(n)=A078367(n+1), n >= 0. This is the m=19 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..18 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364 and A077412. The m=1..3 (signed) sequences are A049347, A056594, A010892. For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 17's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011 For n >= 2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,16}. - Milan Janjic, Jan 23 2015 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..800 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=17, q=-1. Tanya Khovanova, Recursive Sequences W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=19. Index entries for linear recurrences with constant coefficients, signature (17,-1). FORMULA a(n) = 17*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1. a(n) = S(2*n+1, sqrt(19))/sqrt(19) = S(n, 17), where S(n, x) = U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310. a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (17+sqrt(285))/2 and am = (17-sqrt(285))/2. G.f.: 1/(1-17*x+x^2). a(n) = Sum_{k=0..n} A101950(n,k)*16^k. - Philippe Deléham, Feb 10 2012 Product {n >= 0} (1 + 1/a(n)) = (1/15)*(15 + sqrt(285)). - Peter Bala, Dec 23 2012 Product {n >= 1} (1 - 1/a(n)) = (1/34)*(15 + sqrt(285)). - Peter Bala, Dec 23 2012 For n >= 1, a(n) = U(n-1,13/2), where U(k,x) represents Chebyshev polynomial of the second order. - Milan Janjic, Jan 23 2015 a(n) = sqrt((A078367(n+1)^2 - 4)/285), n>=0, (Pell equation d=285, +4). MATHEMATICA CoefficientList[Series[1/(1 - 17 x + x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Dec 24 2012 *) LinearRecurrence[{17, -1}, {1, 17}, 20] (* Harvey P. Dale, Aug 02 2018 *) PROG (Sage) [lucas_number1(n, 17, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008 (MAGMA) I:=[1, 17, 288]; [n le 3 select I[n] else 17*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012 (PARI) my(x='x+O('x^20)); Vec(1/(1-17*x+x^2)) \\ G. C. Greubel, May 25 2019 (GAP) a:=[1, 17, 288];; for n in [4..20] do a[n]:=17*a[n-1]-a[n-2]; od; a; # G. C. Greubel, May 25 2019 CROSSREFS Cf. A077428, A078355 (Pell +4 equations). Cf. A078367. Sequence in context: A196743 A196901 A171322 * A309803 A045607 A045606 Adjacent sequences:  A078363 A078364 A078365 * A078367 A078368 A078369 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Nov 29 2002 STATUS approved

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Last modified October 16 10:55 EDT 2019. Contains 328056 sequences. (Running on oeis4.)