A094649 An accelerator sequence for Catalan's constant.

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

Michael De Vlieger, Table of n, a(n) for n = 0..3649

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.

L. E. Jeffery, Unit-primitive matrices

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)

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

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

Cf. A000032, A094648, A094650.

Sequence in context: A286387 A363974 A342633 * A344970 A135857 A156558

Adjacent sequences: A094646 A094647 A094648 * A094650 A094651 A094652

Keyword

easy,nonn

Author

Paul Barry, May 18 2004

Status

approved