A218655 The Berndt-type sequence number 10 for the argument 2*Pi/13.

2, 4, 13, -176, -786, -3452, 54483, 237722, 1037569, -16329149, -71279530, -311145495, 4897036897, 21376227709, 93310132523, -1468582101731, -6410560285891, -27982966049682, 440416091468393, 1922476035761802, 8391868916275609

Offset: 0

Author

Roman Witula, Nov 04 2012

Keywords

• sign

Comments

A211988(n) + a(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 could be deduced 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).

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

Examples

			Let us put b(n) = A211988(n) + a(n)*sqrt(13). Then we get b(0) = 2*sqrt(13), b(1) = -6 + 4*sqrt(13), b(2) = -37 + 13*sqrt(13), b(3) = 676 - 176*sqrt(13), b(4) = 2882 - 786*sqrt(13), b(5) = 12502 - 3452*sqrt(13).
		

References

• R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, 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

• R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.

Crossrefs

Cf. A211988, A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.