This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A216540 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). 9
 0, 0, -1, -8, -45, -221, -1014, -4472, -19227, -81224, -338767, -1399320, -5736705, -23377770, -94804944, -382930847, -1541565610, -6188513994, -24784429501, -99058333803, -395227906723, -1574536914951, -6264614281978, -24896955293210, -98848880984490 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS a(n) is equal to the rational part (with respect of the field Q(sqrt(13))) of the product sqrt(2(13-3*sqrt(13)))*X(2*n-1)/13, 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 5 for the argument 2Pi/13 defined by the relation A161905(n) + a(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. It follows that s(2) + s(5) + s(6) = s(1)*s(3)*s(4) = sqrt((13 + 3*sqrt(13))/2) and s(2)*s(5)*s(6) = s(1) + s(3) - s(4) =  sqrt((13 - 3*sqrt(13))/2). a(n) is equal to the negated rational part (with respect of the field Q(sqrt(13))) of the product sqrt(2(13+3*sqrt(13)))*Y(2*n-1)/13, 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. Moreover we have A161905(n) - a(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 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 Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13). FORMULA G.f.: -x^3*(2*x-1)*(3*x-1)/(13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). [Colin Barker, Sep 23 2012] EXAMPLE We note that: s(2)^3 + s(5)^3 + s(6)^3 = 2*(s(2) + s(5) + s(6)),  s(2)^5 + s(5)^5 + s(6)^5 = 5* sqrt((13 + 3*sqrt(13))/2) - sqrt((13 - 3*sqrt(13))/2). MATHEMATICA LinearRecurrence[{13, -65, 156, -182, 91, -13}, {0, 0, -1, -8, -45, -221}, 30] CROSSREFS Cf. A216605, A216486, A216597, A216508, A161905, A216801. Sequence in context: A026015 A002696 A016208 * A026852 A317405 A110609 Adjacent sequences:  A216537 A216538 A216539 * A216541 A216542 A216543 KEYWORD sign,easy AUTHOR Roman Witula, Sep 12 2012 EXTENSIONS Better name from Joerg Arndt, Sep 17 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)