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A236337
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Expansion of (2 - x) / ((1 - x)^2 * (1 - x^3)) in powers of x.
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2
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2, 3, 4, 7, 9, 11, 15, 18, 21, 26, 30, 34, 40, 45, 50, 57, 63, 69, 77, 84, 91, 100, 108, 116, 126, 135, 144, 155, 165, 175, 187, 198, 209, 222, 234, 246, 260, 273, 286, 301, 315, 329, 345, 360, 375, 392, 408, 424, 442, 459, 476, 495, 513, 531, 551, 570, 589
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OFFSET
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0,1
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COMMENTS
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The sequence is a quasi-polynomial sequence.
Given a sequence of Laurent polynomials defined by b(n) = (b(n-2)^2 - b(n-1)*b(n-3) * 2/x) / b(n-4), b(-4) = x, b(-3) = b(-2) = -b(-1) = 1. Then the denominator of b(n) is x^a(n).
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LINKS
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FORMULA
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0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z.
G.f.: (2 - x) / ((1 - x)^2 * (1 - x^3)).
Second difference is period 3 sequence [0, 2, -1, ...].
a(n) = 2*a(n-3) - a(n-6) + 3 = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
a(3*n) = (1/2)*(n + 1)*(3*n + 4); a(3*n+1) = (1/2)*(n + 1)*(3*n + 6); a(3*n+2) = (1/2)*(n + 1)*(3*n + 8). - Peter Bala, Feb 11 2019
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EXAMPLE
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G.f. = 2 + 3*x + 4*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 15*x^6 + 18*x^7 + 21*x^8 + ...
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MATHEMATICA
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CoefficientList[Series[(2-x)/((1-x)^2*(1-x^3)), {x, 0, 60}], x] (* Vaclav Kotesovec, Jan 22 2014 *)
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PROG
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(PARI) {a(n) = ((n+1) * (n+6) + [6, 4, 0][n%3 + 1]) / 6};
(PARI) {a(n) = if( n<0, polcoeff( x^4 * (-1 + 2*x) / ((1 - x)^2 * (1 - x^3)) + x * O(x^-n), -n), polcoeff( (2 - x) / ((1 - x)^2 * (1 - x^3)) + x * O(x^n), n))};
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-x)/((1-x)^2*(1-x^3)))); // G. C. Greubel, Aug 07 2018
(Sage) ((2-x)/((1-x)^2*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 11 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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