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A268742 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). 2

%I

%S 0,1,2,20,30,31,44,98,120,121,146,236,270,271,308,434,480,481,530,692,

%T 750,751,812,1010,1080,1081,1154,1388,1470,1471,1556,1826,1920,1921,

%U 2018,2324,2430,2431,2540,2882,3000,3001,3122,3500,3630,3631,3764,4178,4320,4321

%N 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).

%C The sequence lists all m, in increasing order, such that floor(m/2) + floor(m/3) is a square.

%H Bruno Berselli, <a href="/A268742/b268742.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).

%F 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).

%F 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:

%F a(4*k) = 30*k^2;

%F a(4*k+1) = 30*k^2 + 1;

%F a(4*k+2) = 30*k^2 + 12*k + 2;

%F a(4*k+3) = 30*k^2 + 48*k + 20.

%t 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]

%o (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()

%o (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)))

%o (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);

%o (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)));

%Y Cf. A010761.

%Y Cf. A268251: nonnegative m for which floor(m/2)*floor(m/3) is a square.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Feb 12 2016

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Last modified May 25 18:25 EDT 2022. Contains 354071 sequences. (Running on oeis4.)