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A216861 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), with initial terms 0, -2, -9, -44, -215, -1001.

%I A216861 #33 Feb 08 2025 14:22:20
%S A216861 0,-2,-9,-44,-215,-1001,-4446,-19058,-79677,-327418,-1329601,-5355272,
%T A216861 -21446945,-85548138,-340268656,-1350664731,-5353389340,-21195056584,
%U A216861 -83846301409,-331483318257,-1309872510973,-5174049465897,-20431456722794,-80660347594658
%N A216861 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), with initial terms 0, -2, -9, -44, -215, -1001.
%C A216861 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.
%C A216861 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).
%D A216861 Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).
%H A216861 Paolo Xausa, <a href="/A216861/b216861.txt">Table of n, a(n) for n = 1..1000</a>
%H A216861 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (13,-65,156,-182,91,-13).
%F A216861 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
%e A216861 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).
%t A216861 LinearRecurrence[{13, -65, 156, -182, 91, -13}, {0, -2, -9, -44, -215, -1001}, 25] (* _Paolo Xausa_, Feb 23 2024 *)
%Y A216861 Cf. A216905, A216801, A216540.
%K A216861 sign,easy
%O A216861 1,2
%A A216861 _Roman Witula_, Sep 18 2012
%E A216861 Name clarified by _Robert C. Lyons_, Feb 08 2025