A085903 G.f.: (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

1,3

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

Floretion Algebra Multiplication Program, FAMP Code: 4kbasekseq[A*B] with A = + .25'i + .25'j + .25'k + .25i' + .25j' + .25k' + .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' + .25e and B = + .5'i + .5i' + 'ii' + e - Creighton Dement, May 19 2005
(PARI) a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ (Wasserman)
(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

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

Status

approved