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A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

%I A215919 #16 Feb 17 2024 06:27:35
%S A215919 0,-3,12,-36,105,-303,873,-2514,7239,-20844,60018,-172815,497601,
%T A215919 -1432785,4125540,-11879019,34204272,-98487276,283582809,-816544155,
%U A215919 2351145189,-6769852758,19493014119,-56127897168,161613838746,-465348502119,1339917609189,-3858138988821
%N A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.
%C A215919 The Berndt-type sequence number 10 for the argument 2Pi/9 defined by the first trigonometric relation from the section "Formula" below. The sequence a(n) is connected with sequences A215917 and A215885 - see the respective formula.
%C A215919 We have A035045(n)=abs(a(n+1)/3) for every n=0,1,...,5 and A035045(7) + a(7)/3 = 1, A035045(8) - a(8)/3 = 10, A035045(9) + a(9)/3 = 63, and A035045(10) - a(10)/3 = 320 - all these four results-numbers are in A069269.
%D A215919 D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
%D A215919 R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).
%H A215919 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-3,0,1).
%F A215919 a(n) = c(1)*(-c(2))^(-n) + c(2)*(-c(4))^(-n) + c(4)*(-c(1))^(-n), where c(j) := 2*cos(2*Pi*j/9).
%F A215919 a(n) = A215917(n+1) + A215917(n) - 2*(-1)^n*A215885(n).
%F A215919 G.f.: -3*x*(1-x)/(1+3*x-x^3).
%e A215919 We have a(2)=-4*a(1), a(3)=-3*a(2), a(6)/a(3) = -24.25, and a(9) = 579*a(3).
%t A215919 LinearRecurrence[{-3, 0, 1}, {0, -3, 12}, 50]
%Y A215919 Cf. A215917, A215885, A215664.
%K A215919 sign,easy
%O A215919 0,2
%A A215919 _Roman Witula_, Aug 27 2012