This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A215636 #27 Feb 19 2025 03:43:10
%S A215636 0,0,0,-3,24,-135,660,-3003,13104,-55689,232500,-958617,3916440,
%T A215636 -15890355,64127700,-257698347,1032023136,-4121456625,16421256420,
%U A215636 -65301500577,259259758056,-1027901275131,4070632899300,-16104283594083,63657906293520,-251447560563465,992593021410900
%N A215636 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)=a(1)=a(2)=0, a(3)=-3, a(4)=24, a(5)=-135.
%C A215636 The Berndt-type sequence number 4 for the argument 2*Pi/9 defined by the relation: X(n) = b(n) + a(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) = A215635(n) (see also section "Example" below). For more details - see comments to A215635, A215634 and Witula-Slota's reference.
%H A215636 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 A215636 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (-12,-54,-112,-105,-36,-2).
%F A215636 G.f.: (-3*x^3-12*x^4-9*x^5)/(1+12*x+54*x^2+112*x^3+105*x^4+36*x^5+2*x^6).
%e A215636 We have X(1)=-6, X(2)=18 and X(3)=-60-3*sqrt(2), which implies the equality: (cos(Pi/24))^6 + (cos(7*Pi/24))^6 + (cos(3*Pi/8))^6 = (60+3*sqrt(2))/64.
%t A215636 LinearRecurrence[{-12,-54,-112,-105,-36,-2}, {0,0,0,-3,24,-135}, 50]
%Y A215636 Cf. A215455, A215634, A215635, A215664, A215885, A215665, A215666, A215829, A215831, A215917, A215919, A215945, A216034, A215948, A216757.
%K A215636 sign,easy
%O A215636 0,4
%A A215636 _Roman Witula_, Aug 18 2012