A217549 The Berndt-type sequence number 8 for the argument 2*Pi/13.

0, -1, 21, 85, 365, -5707, -24935, -108872, 1713705, 7480420, 32652893, -513913649, -2243303605, -9792325686, 154118686736, 672748988550, 2936640671285, -46218967738367, -201752069488280, -880675175822422, 13860700755359325, 60503840705600655, 264107479466296733

Offset: 0

Author

Roman Witula, Oct 06 2012

Keywords

• sign

Comments

a(n) is defined by the relation A217548(n) + a(n)*sqrt(13)= A(2*n)*2*13^(floor((n+1)/3)/2), where 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, A(1) = sqrt((13-3*sqrt(13))/26).

However the basic sequence 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).

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

Examples

			We have A(1) = A(-1) = sqrt((13-3*sqrt(13))/2), A(2) = (7-sqrt(13))/2, A(3) = (2*sqrt(13)-3)*sqrt((13-3*sqrt(13))/26), A(4) = (21-5*sqrt(13))/2, A(5) = ((13*sqrt(13)-37)/2)*sqrt((13-3*sqrt(13))/26), 2*sqrt(13)*A(6)  = -295 + 85*sqrt(13), and 2*sqrt(13)*(A(6) - 4*A(4)) + 2*A(2) = -28. Furthermore it can be verified that  -a(5)/13 - a(4) - a(3) = A217548(5)/13 + A217548(4) + A217548(3) = -11.
		

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. A216605, A216486, A216597, A216508, A216540, A161905, A217548, A216450, A216801, A216861.