OFFSET
0,4
COMMENTS
The Berndt-type sequence number 2 for the argument 2*Pi/13 defined by the following relation: A216605(n) + a(n)*sqrt(13) = A(n) = 2*(c(1)^n + c(3)^n + c(4)^n), where c(j) := 2*cos(2*Pi*j/13), j=1..6. The numbers a(n), n=0,1,..., are all positive integers. We note that we also have A216605(n) - a(n)*sqrt(13) = B(n) = 2*(c(2)^n + c(5)^n + c(6)^n) and the following recurrence relation holds: B(n) = -((sqrt(13)+1)/2)*B(n-1) + B(n-2) + ((3+sqrt(13))/2)*B(n-3), with B(0) = 6, B(1) = -sqrt(13) - 1, and B(2) = 11 + sqrt(13).
We note that the sums a(2*n+1) + a(2*n+2) are nonnegative only for n = 0..5.
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
Andrew Howroyd, Table of n, a(n) for n = 0..500
R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
Index entries for linear recurrences with constant coefficients, signature (-1,5,4,-6,-3,1).
FORMULA
G.f.: x*(1 - 2*x^2 + 2*x^3 + x^4)/(1 + x - 5*x^2 - 4*x^3 + 6*x^4 + 3*x^5 - x^6).
a(n) = - a(n-1) + 5*a(n-2) + 4*a(n-3) - 6*a(n-4) - 3*a(n-5) + a(n-6), which from the respective polynomial-type formula follows given by Witula in section "Formula" in A216605.
EXAMPLE
We have a(5) + a(6) + a(4) + a(2) = a(7) + a(8) + a(6) + a(2) = a(9) + a(5) + a(1) + a(10) + a(8) = 0 and
a(6) + a(9) + a(10) = a(11) + a(12) = 12.
Moreover, the following relations hold: A(3) = 4*A(1), B(3) = 4*B(1), A(5) = 4*A(3) + 2*sqrt(13), B(5) = 4*B(3)-2*sqrt(13), A(7) = 4*A(5) + 8*sqrt(13), B(7) = 4*B(5)-8*sqrt(13), A(4) = 3*A(2) - 2, B(4) = 3*B(2) + 2, 6 + A(6) = 3*A(4) + A(2), and A(8) - 3*A(6) = 25 - A(5)/2.
MATHEMATICA
LinearRecurrence[{-1, 5, 4, -6, -3, 1}, {0, 1, -1, 4, -3, 14}, 30]
PROG
(PARI) concat([0], Vec((1-2*x^2+2*x^3+x^4)/(1+x-5*x^2-4*x^3+6*x^4+3*x^5-x^6) + O(x^30))) \\ Andrew Howroyd, Feb 25 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roman Witula, Sep 11 2012
STATUS
approved