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A215635 a(n) = - 12*a(n-1) - 54*a(n-2) - 112*a(n-3) - 105*a(n-4) -36*a(n-5) - 2*a(n-6), with a(0)=3, a(1)=-6, a(2)=18, a(3)=-60, a(4)=210, a(5)=-756.

%I A215635 #28 Feb 19 2025 02:45:40
%S A215635 3,-6,18,-60,210,-756,2772,-10296,38610,-145860,554268,-2116296,
%T A215635 8112462,-31201644,120347532,-465328200,1803025410,-6999149124,
%U A215635 27213719148,-105960069864,413078158350,-1612098272460,6297409350492,-24620247483624,96324799842498,-377102656201956,1477141800784668
%N A215635 a(n) = - 12*a(n-1) - 54*a(n-2) - 112*a(n-3) - 105*a(n-4) -36*a(n-5) - 2*a(n-6), with a(0)=3, a(1)=-6, a(2)=18, a(3)=-60, a(4)=210, a(5)=-756.
%C A215635 The Berndt-type sequence number 3 for the argument 2*Pi/9 defined by the relation: X(n) = a(n) + b(n)*sqrt(2), where X(n) := ((cos(Pi/24))^(2*n) + (cos(7*Pi/24))^(2*n) + (cos(3*Pi/8))^(2*n))*(-4)^n. We have b(n) = A215636(n).
%C A215635 We note that above formula is the Binet form of the following recurrence sequence: X(n+3) + 6*X(n+2) + 9*X(n+1) + (2 + sqrt(2))*X(n) = 0, which is a special type of the sequence X(n)=X(n;g) defined in the comments to A215634 for g:=Pi/24. The sequences a(n) and b(n) satisfy the following system of recurrence equations: a(n) = -b(n+3)-6*b(n+2)-9*b(n+1)-2*b(n), 2*b(n) =  -a(n+3)-6*a(n+2)-9*a(n+1)-2*a(n).
%C A215635 There exists an amazing relation: (-1)^n*a(n)=3*A000984(n) for every n=0,1,...,11 and 3*A000984(12)-a(12)=6.
%H A215635 Roman Witula and D. Slota, <a href="http://dx.doi.org/10.1016/j.jmaa.2005.12.020">On modified Chebyshev polynomials</a>, J. Math. Anal. Appl., 324 (2006), 321-343.
%H A215635 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (-12,-54,-112,-105,-36,-2).
%F A215635 G.f.: (3+30*x+108*x^2+168*x^3+105*x^4+18*x^5) / (1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6).
%t A215635 LinearRecurrence[{-12,-54,-112,-105,-36,-2}, {3,-6,18,-60,210,-756}, 50]
%o A215635 (PARI) Vec((3+30*x+108*x^2+168*x^3+105*x^4+18*x^5) /(1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%Y A215635 Cf. A215455, A215634, A215636, A215007, A214699, A214683.
%K A215635 sign,easy
%O A215635 0,1
%A A215635 _Roman Witula_, Aug 18 2012