A085903 Expansion of (1 + 2*x^2)/((1 + x)*(1 - 2*x)*(1 - 2*x^2)).
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
Offset: 1
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).
Programs
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Magma
[Round((Sqrt(2)^n - 1)*(Sqrt(2)^n - (-1)^n)): n in [1..40]]; // Vincenzo Librandi, Apr 28 2014
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Maple
seq(simplify((sqrt(2)^n - 1)*(sqrt(2)^n - (-1)^n)), n = 1..30); # Peter Bala, Apr 27 2014
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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 *) -
PARI
a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ David Wasserman, Feb 10 2005
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Python
def A085903(n): return (1<
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
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
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
Comments