login
A094649
An accelerator sequence for Catalan's constant.
6
4, 1, 7, 4, 19, 16, 58, 64, 187, 247, 622, 925, 2110, 3394, 7252, 12289, 25147, 44116, 87727, 157492, 307294, 560200, 1079371, 1987891, 3798310, 7043041, 13382818, 24927430, 47191492, 88165105, 166501903, 311686804, 587670811, 1101562312
OFFSET
0,1
COMMENTS
From L. Edson Jeffery, Apr 03 2011: (Start)
Let U be the unit-primitive matrix (see [Jeffery])
U = U_(9,1) =
(0 1 0 0)
(1 0 1 0)
(0 1 0 1)
(0 0 1 1).
Then a(n) = Trace(U^n). (End)
a(n)==1 (mod 3), a(3*n+1)==1 (mod 9). - Roman Witula, Sep 14 2012
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, sequence a(n) with l=9.
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.
Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019.
Q. Wang, On generalized Lucas sequences, Contemp. Math. 531 (2010) 127-141, Table 2 (k=4)
FORMULA
G.f.: ( 4-3*x-6*x^2+2*x^3 ) / ( (x-1)*(x^3+3*x^2-1) )
a(n) = 1+(2*cos(Pi/9))^n+(-2*sin(Pi/18))^n+(-2*cos(2*Pi/9))^n.
a(n) = 2^n*Sum_{k=1..4} cos((2*k-1)*Pi/9)^n. - L. Edson Jeffery, Apr 03 2011
a(n) = 1 + (-1)^n*A215664(n), which is compatible with the last two formulas above. - Roman Witula, Sep 14 2012
a(n) = 3*a(n-2) + a(n-3) - 3, with a(0)=4, a(1)=1, and a(2)=7. - Roman Witula, Sep 14 2012
EXAMPLE
We have a(0)+a(3)=a(1)+a(2)=8, a(3)+a(4)=a(2)+a(5)=23, and a(7)+a(8)=a(9)+a(3)=247. - Roman Witula, Sep 14 2012
MATHEMATICA
LinearRecurrence[{1, 3, -2, -1}, {4, 1, 7, 4}, 34] (* Jean-François Alcover, Sep 21 2017 *)
PROG
(PARI) Vec((4-3*x-6*x^2+2*x^3)/(1-x-3*x^2+2*x^3+x^4)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 18 2004
STATUS
approved