OFFSET
0,1
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
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Roman Witula, Aug 25 2012
STATUS
approved