A094650 An accelerator sequence for Catalan's constant.
5, -1, 9, -4, 25, -16, 78, -64, 257, -256, 874, -1013, 3034, -3953, 10684, -15229, 38017, -58056, 136338, -219508, 491870, -824737, 1782735, -3083887, 6484514, -11489516, 23652443, -42688039, 86459608, -158270401, 316576903, -585868009, 1160673633
Offset: 0
Links
- A. Akbary and Q. Wang, On some permutation polynomials over finite fields, International Journal of Mathematics and Mathematical Sciences, 2005:16 (2005) 2631-2640.
- A. Akbary and Q. Wang, A generalized Lucas sequence and permutation binomials, Proceeding of the American Mathematical Society, 134 (1) (2006), 15-22.
- David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, The Ramanujan Journal, Vol. 3, Issue 2, 1999, pp. 159-173
- David M. Bradley, A Class of Series Acceleration Formulae for Catalan's Constant, arXiv:0706.0356 [math.CA], 2007.
- Russell A. Gordon, Lucas Type Sequences and Sums of Binomial Coefficients, Integers (2023) Vol 23, Art. No. A84. See p. 21.
- Index entries for linear recurrences with constant coefficients, signature (-1,4,3,-3,-1).
Programs
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Mathematica
LinearRecurrence[{-1, 4, 3, -3, -1}, {5, -1, 9, -4, 25}, 33] (* Jean-François Alcover, Sep 21 2017 *) -
PARI
Vec((5+4*x-12*x^2-6*x^3+3*x^4)/(1+x-4*x^2-3*x^3+3*x^4+x^5) + O(x^40)) \\ Michel Marcus, Jul 25 2015
Formula
G.f.: (5+4x-12x^2-6x^3+3x^4)/(1+x-4x^2-3x^3+3x^4+x^5).
a(n) = (2*cos(2*Pi/11))^n + (-2*cos(Pi/11))^n + (-2*sin(5*Pi/22))^n +(2*sin(3*Pi/22))^n + (-2*sin(Pi/22))^n.