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A321703
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a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), a(0) = 1, a(1) = 1, a(2) = 5.
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2
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1, 1, 5, 20, 81, 328, 1328, 5377, 21771, 88149, 356908, 1445091, 5851054, 23690434, 95920609, 388374617, 1572496721, 6366909240, 25779089221, 104377401344, 422615470156, 1711135105065, 6928244597163, 28051889681905, 113579782539432, 459875150943079, 1861996472668870, 7539069804318358, 30525070454573633, 123593487053663201, 500419812783602493
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OFFSET
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0,3
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COMMENTS
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In general, let {X,Y,Z} be the roots of the cubic equation x^3 + ax^2 + xt + c = 0 where {a, b, c} are integers.
Let {u, v, w} be three numbers such that {u + v + w, u*X + v*Y + w*Z, u*X^2 + v*Y^2 + w*Z^2} are integers.
Then {p(n) = u*X^n + v*Y^n + w*Z^n | n = 0, 1, 2, ...} is an integer sequence with the recurrence relation: p(n) = -a*p(n-1) - b*p(n-2) - c*p(n-3).
Let k = Pi/7.
Let X = (sin(4k)*sin(8k))/(sin(2k)*sin(2k)),
Y = (sin(8k)*sin(2k))/(sin(4k)*sin(4k)),
Z = (sin(2k)*sin(4k))/(sin(8k)*sin(8k)).
Then {X,Y,Z} are the roots of the cubic equation x^3 - 3*x^2 - 4*x - 1 = 0.
This sequence: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k))).
A122600: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k)), 1/(sqrt(7)*tan(8k))).
A321715: (a, b, c) = (3, 4, 1), (u, v, w) = (1/(sqrt(7)*tan(8k)), 1/(sqrt(7)*tan(2k)), 1/(sqrt(7)*tan(4k))).
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LINKS
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FORMULA
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G.f.: (-1 + 2*x + 2*x^2)/(-1 + 3*x + 4*x^2 + x^3). - Stefano Spezia, Jan 14 2019
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MATHEMATICA
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CoefficientList[Series[(-1 + 2 x + 2 x^2)/(-1 + 3 x + 4 x^2 + x^3), {x, 0, 50}], x] (* Stefano Spezia, Jan 14 2019 *)
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PROG
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(PARI) Vec((1 - 2*x - 2*x^2) / (1 - 3*x - 4*x^2 - x^3) + O(x^30)) \\ Colin Barker, Jan 15 2019
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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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