A215817 a(n) is the rational part of A(n) = (6-sqrt(7))*A(n-1) - (12-4*sqrt(7))*A(n-2) + (8-3*sqrt(7))*A(n-3) with A(0)=3, A(1)=6-sqrt(7), A(2)=19-4*sqrt(7).

3, 6, 19, 66, 237, 866, 3202, 11948, 44917, 169914, 646134, 2467988, 9462498, 36398004, 140399901, 542894726, 2103745125, 8167514346, 31762430143, 123704647562, 482435457922, 1883712663668, 7363103647479, 28809291337986, 112820819490970, 442175629583316

Offset: 0

Author

Roman Witula, Aug 25 2012

Keywords

• nonn

Comments

The Berndt-type sequence number 14 for the argument 2Pi/7 defined by requiring a(n) to be the rational part of the trigonometric sum A(n) := c(1)^(2*n) + c(2)^(2*n) + c(4)^(2*n), where c(j) := 2*cos(Pi/4 + 2*Pi*j/7) = 2*cos((7+8*j)*Pi/28). We note that (A(n)-a(n))/sqrt(7) = A215877(n) are all integers. We have A(n)=2^n*O(n;i/2), where O(n;d) denote the big omega function with index n for the argument d in C defined in comments to A215794 (see also Witula-Slota's paper - Section 6). From the respective recurrence relation for this function we generate the title recurrence for A(n).

Links

• Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
• Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5

Crossrefs

Cf. A215493, A215494, A215143, A215510, A094429, A215794.

Formula

a(n) = rational part of c(1)^(2n) + c(2)^(2n) + c(4)^(2n) = (1-s(1))^n + (1-s(2))^n + (1-s(4))^n, where c(j) := 2*cos((7+8*j)/28) and s(j) := sin(2*Pi*j/7).

Empirical g.f.: -(2*x-1)*(6*x^4 -40*x^3 +58*x^2 -24*x +3) / (x^6 -24*x^5 +86*x^4 -104*x^3 +53*x^2 -12*x +1). - Colin Barker, Jun 01 2013