OFFSET
1,2
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 sequence X(n) is defined in almost the same way as sequence Y(n) from the comments to A161905. The only difference is in the initial condition X(2) = -Y(2).
REFERENCES
Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-65,156,-182,91,-13).
FORMULA
G.f.: -x^2*(26*x^4-84*x^3+57*x^2-17*x+2) / (13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - Colin Barker, Jun 01 2013
EXAMPLE
We have a(3)-5*a(2)=a(4)-5a(3)=1, a(5)-5*a(4)=5, and 19000 + a(8) = a(4) + 2*a(3) - 2*a(2).
MATHEMATICA
LinearRecurrence[{13, -65, 156, -182, 91, -13}, {0, -2, -9, -44, -215, -1001}, 25] (* Paolo Xausa, Feb 23 2024 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roman Witula, Sep 18 2012
STATUS
approved