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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. 13
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 (list; graph; refs; listen; history; text; internal format)
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
R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
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
Sequence in context: A069730 A072605 A330344 * A030953 A030811 A030917
KEYWORD
sign,easy,changed
AUTHOR
Roman Witula, Sep 12 2012
EXTENSIONS
Better name from Joerg Arndt, Sep 17 2012
STATUS
approved

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Last modified February 21 03:13 EST 2024. Contains 370219 sequences. (Running on oeis4.)