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 A356563 #10 Aug 13 2022 06:24:46
%S A356563 3,2,6,20,50,132,354,940,2498,6644,17666,46972,124898,332100,883042,
%T A356563 2347980,6243202,16600468,44140098,117367068,312075170,829797604,
%U A356563 2206404514,5866756972,15599513666,41478593332,110290214274
%N A356563 Sums of powers of roots of x^3 - 2*x^2 - x - 2.
%C A356563 The three roots of x^3 - 2*x^2 - x - 2 are c1=2.65896708... = A348909+1, c2=-0.32948354... + 0.80225455...*i, and c3=-0.32948354... - 0.80225455...*i.
%C A356563 a(n) can also be determined by Vieta's formulas and Newton's identities. For example, a(3) by definition is c1^3 + c2^3 + c3^3, and from Newton's identities this equals e1^3 - 3*e1*e2 + 3*e3 for e1, e2, e3 the elementary symmetric polynomials of x^3 - x^2 - x - 3. From Vieta's formulas we have e1 = 2, e2 = -1, and e3 = 2, giving us e1^3 - 3*e1*e2 + 3*e3 = 8 + 6 + 6 = 20, as expected.
%H A356563 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,2).
%F A356563 a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3).
%F A356563 G.f.: (3 - 4 x - x^2)/(1 - 2 x - x^2 - 2 x^3).
%F A356563 2*a(n) = 7*b(n) - b(n+1) for b(n) = A077996(n).
%e A356563 For n=3, a(3) = (2.65896708...)^3 + (-0.32948354... + 0.80225455...*i)^3 + (-0.32948354... - 0.80225455...*i)^3 = 20.
%t A356563 LinearRecurrence[{2, 1, 2}, {3, 2, 6}, 30]
%Y A356563 Cf. A077996, A348909 (c1-1).
%K A356563 nonn,easy
%O A356563 0,1
%A A356563 _Greg Dresden_ and _Hanzhang Fang_, Aug 12 2022