

A215817


a(n) is the rational part of A(n) = (6sqrt(7))*A(n1)  (124*sqrt(7))*A(n2) + (83*sqrt(7))*A(n3) with A(0)=3, A(1)=6sqrt(7), A(2)=194*sqrt(7).


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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

The Berndttype 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 WitulaSlota's paper  Section 6). From the respective recurrence relation for this function we generate the title recurrence for A(n).


LINKS

Table of n, a(n) for n=0..25.
Roman Witula and Damian Slota, New RamanujanType Formulas and QuasiFibonacci 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) = (1s(1))^n + (1s(2))^n + (1s(4))^n, where c(j) := 2*cos((7+8*j)/28) and s(j) := sin(2*Pi*j/7).
Empirical g.f.: (2*x1)*(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

Cf. A215493, A215494, A215143, A215510, A094429, A215794.
Sequence in context: A186022 A058818 A184937 * A269306 A326317 A306522
Adjacent sequences: A215814 A215815 A215816 * A215818 A215819 A215820


KEYWORD

nonn


AUTHOR

Roman Witula, Aug 25 2012


STATUS

approved



