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A356463 Sum of powers of roots of x^3 - 4*x^2 + x + 1.

%I A356463 #22 Sep 13 2022 09:36:48
%S A356463 3,4,14,49,178,649,2369,8649,31578,115294,420949,1536924,5611453,
%T A356463 20487939,74803379,273114124,997165178,3640743209,13292693534,
%U A356463 48532865749,177198026253,646966545729,2362135290914
%N A356463 Sum of powers of roots of x^3 - 4*x^2 + x + 1.
%C A356463 The three roots of x^3 - 4*x^2 + x + 1 are c1 = 1 + 2*cos(Pi/13) + 2*cos(5*Pi/13), c2 = 1 + 2*cos(3*Pi/13) + 2*cos(11*Pi/13), and c3 = 1 + 2*cos(7*Pi/13) + 2*cos(9*Pi/13), so our entries are a(n) = c1^n + c2^n + c3^n.
%H A356463 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-1).
%F A356463 G.f.: (3 - 8*x + x^2)/(1 - 4*x + x^2 + x^3).
%F A356463 a(n) = 4*a(n-1) - a(n-2) - a(n-3).
%F A356463 a(n) = 3*b(n-1) + 5*b(n-2) + b(n-3) for b(n) = A052941(n).
%F A356463 a(n) = round(c1^n), c1 per the comment, n >= 3. - _Bill McEachen_, Sep 12 2022
%t A356463 LinearRecurrence[{4, -1, -1}, {3, 4, 14},30]
%o A356463 (PARI) a(n) = ([0,1,0;0,0,1;-1,-1,4]^n * [3;4;14])[1,1] \\ _Jianing Song_, Aug 09 2022
%o A356463 (PARI) polsym(x^3 - 4*x^2 + x + 1, 33) \\ _Joerg Arndt_, Sep 13 2022
%Y A356463 Cf. A052941.
%K A356463 nonn,easy
%O A356463 0,1
%A A356463 _Greg Dresden_ and _Ding Hao_, Aug 08 2022