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

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 16 2003

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A083420, A028400, A062510, A088037, A107663.

Cf. A060867. A100047.

Programs

Magma

[Round((Sqrt(2)^n - 1)*(Sqrt(2)^n - (-1)^n)): n in [1..40]]; // Vincenzo Librandi, Apr 28 2014

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 *)

PARI

a(n) = polresultant(x^n - 1, 2*x^2 - 1) \\ David Wasserman, Feb 10 2005

Python

def A085903(n): return (1<>1))-1)**2 # Chai Wah Wu, Jun 19 2024

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

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