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 A085903 G.f.: (1 + 2*x^2)/((1 + x)*(1 - 2*x)*(1 - 2*x^2)). 3
 1, 1, 7, 9, 31, 49, 127, 225, 511, 961, 2047, 3969, 8191, 16129, 32767, 65025, 131071, 261121, 524287, 1046529, 2097151, 4190209, 8388607, 16769025, 33554431, 67092481, 134217727, 268402689, 536870911, 1073676289, 2147483647 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Resultant of the polynomial x^n - 1 and the Chebyshev polynomial of the first kind T_2(x). This sequence is the case P1 = 1, P2 = 0, Q = -2 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 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 entries for linear recurrences with constant coefficients, signature (1,4,-2,-4). FORMULA a(2*n) = 2*4^n - 1, a(2*n + 1) = (2^n - 1)^2; interlaces A083420 with A060867 (squares of Mersenne numbers A000225). - Creighton Dement, May 19 2005 A107663(2*n) = a(2*n) = A083420(n). - Creighton Dement, May 19 2005 From Peter Bala, Apr 27 2014: (Start) a(n) = (sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)). a(n) = Product_{k = 1..n} ( 2 - exp(4*k*Pi*i/n) ). (End) E.g.f.: exp(-x) + exp(2*x) - 2*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Jun 16 2016 MAPLE seq(simplify((sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)), n = 1..30); # Peter Bala, Apr 27 2014 MATHEMATICA CoefficientList[ Series[(1 + 2x^2)/(1 - x - 4x^2 + 2x^3 + 4x^4), {x, 0, 30}], x] (* Robert G. Wilson v, May 04 2013 *) LinearRecurrence[{1, 4, -2, -4}, {1, 1, 7, 9}, 40] (* Harvey P. Dale, Jul 25 2016 *) PROG (PARI) a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ David Wasserman, Feb 10 2005 (Magma) [Round((Sqrt(2)^n - 1)*(Sqrt(2)^n - (-1)^n)): n in [1..40]]; // Vincenzo Librandi, Apr 28 2014 CROSSREFS Cf. A083420, A028400, A062510, A088037, A107663. Cf. A060867. A100047. Sequence in context: A154260 A309199 A186234 * A200180 A147248 A147186 Adjacent sequences: A085900 A085901 A085902 * A085904 A085905 A085906 KEYWORD nonn,easy AUTHOR Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003 EXTENSIONS More terms from David Wasserman, Feb 10 2005 Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007 STATUS approved

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Last modified May 29 07:06 EDT 2024. Contains 372926 sequences. (Running on oeis4.)