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A356463
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Sum of powers of roots of x^3 - 4*x^2 + x + 1.
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2
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3, 4, 14, 49, 178, 649, 2369, 8649, 31578, 115294, 420949, 1536924, 5611453, 20487939, 74803379, 273114124, 997165178, 3640743209, 13292693534, 48532865749, 177198026253, 646966545729, 2362135290914
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: (3 - 8*x + x^2)/(1 - 4*x + x^2 + x^3).
a(n) = 4*a(n-1) - a(n-2) - a(n-3).
a(n) = 3*b(n-1) + 5*b(n-2) + b(n-3) for b(n) = A052941(n).
a(n) = round(c1^n), c1 per the comment, n >= 3. - Bill McEachen, Sep 12 2022
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MATHEMATICA
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LinearRecurrence[{4, -1, -1}, {3, 4, 14}, 30]
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PROG
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(PARI) a(n) = ([0, 1, 0; 0, 0, 1; -1, -1, 4]^n * [3; 4; 14])[1, 1] \\ Jianing Song, Aug 09 2022
(PARI) polsym(x^3 - 4*x^2 + x + 1, 33) \\ Joerg Arndt, Sep 13 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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