A161905 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with a(1)..a(6) as shown.

2, 4, 13, 52, 221, 949, 4056, 17186, 72163, 300482, 1241981, 5100758, 20833813, 84695026, 342920942, 1383646433, 5566235714, 22334785486, 89420529809, 357319721889, 1425447435997, 5678246483273, 22590565547134, 89775857333032, 356428030609222, 1413891596961194, 5604509198580578

Offset

1,1

Comments

a(n) is equal to the rational part (with respect to the field Q(sqrt(13))) of the product sqrt(2*(13-3*sqrt(13))/13)*X(2*n-1), where X(n) = sqrt((13 + 3*sqrt(13))/2)*X(n-1) - sqrt(13)*X(n-2) + sqrt((13 - 3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13 + 3*sqrt(13))/2), and X(2)=(13 - sqrt(13))/2.

The Berndt-type sequence number 6 for the argument 2*Pi/13 defined by the relation a(n) + A216540(n)*sqrt(13) = sqrt(2*(13-3*sqrt(13))/13)*X(2*n-1), where X(n) := s(2)^n + s(5)^n + s(6)^n, and s(j) := 2*sin(2*Pi*j/13), j=1,2,...,6.

We note that all numbers a(n+1)-4*a(n) for n=3,4,..., are divisible by 13. For example we have a(4)=4*a(3), a(5)-4*a(4)=13, a(6)-4*a(5)=5*13, a(7)-4*a(6)=20*13, and a(10)-4*a(9)=70*13^2.

a(n) is also equal to the rational part (with respect to the field Q(sqrt(13))) of the product sqrt(2*(13+3*sqrt(13))/13)*Y(2*n-1), where Y(n) = sqrt((13 - 3*sqrt(13))/2)*Y(n-1) + sqrt(13)*Y(n-2) - sqrt((13 + 3*sqrt(13))/2)*Y(n-3), with Y(0)=3, Y(1)=sqrt((13 - 3*sqrt(13))/2), and Y(2)=(13 + sqrt(13))/2. Let us observe that a(n) - A216540(n)*sqrt(13) = sqrt(2*(13+3*sqrt(13))/13)*Y(2*n-1) and Y(n) = s(1)^n + s(3)^n + s(9)^n (we have s(9) = -s(4)). - Roman Witula, Sep 22 2012

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).

Formula

G.f.: -x*(-2 + 22*x - 91*x^2 + 169*x^3 - 130*x^4 + 26*x^5) / (1 - 13*x + 65*x^2 - 156*x^3 + 182*x^4 - 91*x^5 + 13*x^6). - R. J. Mathar, Sep 18 2012

Example

It can be shown that 4*X(5) - X(7) = sqrt(26*(13+3*sqrt(13))), 4*X(7) - X(9) = 13*(sqrt(13) - 1)*sqrt(2*(13 + 3*sqrt(13)))/4, and 4*X(11) - X(13) = 130*(sqrt(13) - 2)*sqrt(2*(13 + 3*sqrt(13)))/4, which implies
(4*X(7) - X(9))/(4*X(5) - X(7)) = 13*(sqrt(13) - 1) and
(4*X(11) - X(13))/(4*X(7) - X(9)) = 10*(sqrt(13) - 2)/(sqrt(13) - 1) = 5*(11 - sqrt(13))/6.
We also have a(6) - a(3) - a(1) = 4000, a(9) - 2*a(4) - a(3) + 3*a(1) = 300000, and a(11) - a(5) + a(4) - 2*a(2) - a(1) = 5100000.

Mathematica

LinearRecurrence[{13, -65, 156, -182, 91, -13}, {2, 4, 13, 52, 221, 949}, 30]
CoefficientList[Series[(2-22 x+91 x^2-169 x^3+130 x^4-26 x^5)/(1-13 x+ 65 x^2- 156 x^3+182 x^4-91 x^5+13 x^6), {x, 0, 40}], x] (* Harvey P. Dale, Jun 05 2021 *)

Crossrefs

Cf. A216605, A216486, A216597, A216508, A216540.

Sequence in context: A069730 A072605 A330344 * A030953 A030811 A030917

Adjacent sequences: A161902 A161903 A161904 * A161906 A161907 A161908

Keyword

sign,easy

Author

Roman Witula, Sep 12 2012

Extensions

Better name from Joerg Arndt, Sep 17 2012

Status

approved