OFFSET
1,1
COMMENTS
a(n) is equal to the rational part of the number 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.
Let us observe that all numbers of the form a(n)*13^(-floor((n+3)/6)) are integers.
We note that the sequence X(n) is defined "similarly" to sequence Y(n) in the comments to A216540. The only difference between them is in 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*(52*x^5-520*x^4+689*x^3-299*x^2+48*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 4*a(3)=a(4), 4*a(4)=a(5)+a(3). The 3-valuation of a(n) for n=1,...,10 is contained in A167366. Moreover it can be obtained X(7) - 22*X(3) = 4*sqrt(2*(13-3*sqrt(13))), 4*X(5) - X(7) = 2*sqrt(26(13-3*sqrt(13))), and 15*X(5) - X(9) = 20*sqrt(26(13-3*sqrt(13))), which implies (15*X(5) - X(9))/(4*X(5) - X(7)) = 10.
MATHEMATICA
LinearRecurrence[{13, -65, 156, -182, 91, -13}, {2, -22, -117, -468, -1755, -6513}, 25] (* Paolo Xausa, Feb 23 2024 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roman Witula, Sep 17 2012
STATUS
approved