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A211988 The Berndt-type sequence number 9 for the argument 2*Pi/13. 11
0, -6, -37, 676, 2882, 12502, -196209, -856850, -3740697, 58876883, 257003504, 1121852777, -17656510365, -77073076671, -336434457597, 5295048110651, 23113603862267, 100894018986142, -1587942800101489, -6931585922526870, -30257313674299627, 476211413709501353 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) + A218655(n)*sqrt(13)  = A(2*n+1)*13^((1+floor(n/3))/2)*sqrt(2*(13 + 3*sqrt(13))/13), where A(n) is defined below.

The sequence A(n) from the name of a(n) is defined  by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) := 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in Comments to A216508).

It follows that A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).

We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2), sqrt(2*sqrt(13))*(s(1)^(-3) + s(3)^(-3) + s(9)^(-3)) = sqrt((97*sqrt(13)-339), and  s(1)^(-9) + s(3)^(-9) + s(9)^(-9) = (131/13)*sqrt(2834 - 786*sqrt(13)).

The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

REFERENCES

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13 on the occasion of the Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.

R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

LINKS

Table of n, a(n) for n=0..21.

CROSSREFS

Cf. A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.

Sequence in context: A041062 A240324 A283636 * A285934 A083373 A320988

Adjacent sequences:  A211985 A211986 A211987 * A211989 A211990 A211991

KEYWORD

sign

AUTHOR

Roman Witula, Oct 25 2012

STATUS

approved

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Last modified November 21 03:01 EST 2018. Contains 317427 sequences. (Running on oeis4.)