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A268742
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Expansion of x*(1 + x + 18*x^2 + 10*x^3 - x^4 + 11*x^5 + 18*x^6 + 2*x^7) / (1 - x - 2*x^4 + 2*x^5 + x^8 - x^9).
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2
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0, 1, 2, 20, 30, 31, 44, 98, 120, 121, 146, 236, 270, 271, 308, 434, 480, 481, 530, 692, 750, 751, 812, 1010, 1080, 1081, 1154, 1388, 1470, 1471, 1556, 1826, 1920, 1921, 2018, 2324, 2430, 2431, 2540, 2882, 3000, 3001, 3122, 3500, 3630, 3631, 3764, 4178, 4320, 4321
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OFFSET
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0,3
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COMMENTS
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The sequence lists all m, in increasing order, such that floor(m/2) + floor(m/3) is a square.
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LINKS
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Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).
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FORMULA
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G.f.: x*(1 + x + 18*x^2 + 10*x^3 - x^4 + 11*x^5 + 18*x^6 + 2*x^7)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).
a(n) = (30*(n-1)*n + 2*(18*n-3*(-1)^n-11)*(-1)^(n*(n+1)/2) - (6*n+1)*(-1)^n + 13)/16 + 1. Therefore:
a(4*k) = 30*k^2;
a(4*k+1) = 30*k^2 + 1;
a(4*k+2) = 30*k^2 + 12*k + 2;
a(4*k+3) = 30*k^2 + 48*k + 20.
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MATHEMATICA
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CoefficientList[x (1 + x + 18 x^2 + 10 x^3 - x^4 + 11 x^5 + 18 x^6 + 2 x^7)/((1 + x)^2 (1 - x)^3 (1 + x^2)^2) + O[x]^50, x]
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PROG
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(Sage) gf = x*(1 + x + 18*x^2 + 10*x^3 - x^4 + 11*x^5 + 18*x^6 + 2*x^7)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2); taylor(gf, x, 0, 50).list()
(PARI) concat(0, Vec((1 + x+18*x^2 + 10*x^3 - x^4 + 11*x^5 + 18*x^6 + 2*x^7)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2) + O(x^50)))
(Maxima) makelist(coeff(taylor(x*(1 + x + 18*x^2 + 10*x^3 - x^4 + 11*x^5 + 18*x^6 + 2*x^7)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2), x, 0, n), x, n), n, 0, 50);
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1 + x + 18*x^2 + 10*x^3 - x^4 + 11*x^5 + 18*x^6 + 2*x^7)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2)));
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CROSSREFS
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Cf. A010761.
Cf. A268251: nonnegative m for which floor(m/2)*floor(m/3) is a square.
Sequence in context: A136903 A078460 A059208 * A261460 A061471 A112271
Adjacent sequences: A268739 A268740 A268741 * A268743 A268744 A268745
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KEYWORD
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nonn,easy
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AUTHOR
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Bruno Berselli, Feb 12 2016
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STATUS
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approved
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