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%I #30 Aug 08 2017 12:27:07
%S 3,56,1568,53312,1931776,71300096,2645479424,98305622016,
%T 3654656065536,135885355483136,5052615982317568,187873377732526080,
%U 6985794697679601664,259756778648305139712,9658687473893481906176,359144636249686988029952,13354285908291066433372160
%N a(n) = (7*(csc(2*Pi/7))^2)^n + (7*(csc(4*Pi/7))^2)^n + (7*(csc(8*Pi/7))^2)^n.
%H Colin Barker, <a href="/A287396/b287396.txt">Table of n, a(n) for n = 0..600</a>
%H Roman Witula, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Witula/witula17.html">Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7</a>, J. Integer Seq., 12 (2009), Article 09.8.5.
%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.html">Quasi-Fibonacci Numbers of Order 11</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5
%H Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (56,-784,3136).
%F a(n) = x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of x^3 - 56*x^2 + 784* x - 3136, x1 = 7*(csc(2*Pi/7))^2, x2 = 7*(csc(4*Pi/7))^2, x3 = 7*(csc(8*Pi/7))^2.
%F a(n) = 56*a(n-1) - 784*a(n-2) + 3136*a(n-3) for n>2, a(0) = 3, a(1) = 56, a(2) = 1568.
%F G.f.: (3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3). - _Colin Barker_, May 25 2017
%t LinearRecurrence[{56,-784,3136},{3,56,1568},30] (* _Harvey P. Dale_, Aug 08 2017 *)
%o (PARI) Vec((3 - 28*x)*(1 - 28*x) / (1 - 56*x + 784*x^2 - 3136*x^3) + O(x^30)) \\ _Colin Barker_, May 25 2017
%o (PARI) polsym(x^3 - 56*x^2 + 784* x - 3136, 20) \\ _Joerg Arndt_, May 26 2017
%Y Cf. A033304, A094648, A274220, A215076, A274075, A274032, A275195, A215575, A274975.
%K nonn,easy
%O 0,1
%A _Kai Wang_, May 24 2017
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