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 A216801 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). 5
 2, -22, -117, -468, -1755, -6513, -24336, -91988, -351689, -1357408, -5277363, -20625774, -80909257, -318173258, -1253243498, -4941450657, -19495914360, -76945654032, -303737001009, -1199041027587, -4733273752831, -18683644465447, -73743457866962 (list; graph; refs; listen; history; text; internal format)
 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 R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish). LINKS Table of n, a(n) for n=1..23. 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. CROSSREFS Cf. A216540, A161905, A216861. Sequence in context: A281140 A105237 A325948 * A083833 A221697 A292452 Adjacent sequences: A216798 A216799 A216800 * A216802 A216803 A216804 KEYWORD sign,easy AUTHOR Roman Witula, Sep 17 2012 STATUS approved

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Last modified December 10 00:12 EST 2023. Contains 367696 sequences. (Running on oeis4.)