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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
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
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
Sequence in context: A154260 A309199 A186234 * A200180 A147248 A147186
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.)