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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).

%I A268742 #23 Dec 30 2024 19:06:17
%S A268742 0,1,2,20,30,31,44,98,120,121,146,236,270,271,308,434,480,481,530,692,
%T A268742 750,751,812,1010,1080,1081,1154,1388,1470,1471,1556,1826,1920,1921,
%U A268742 2018,2324,2430,2431,2540,2882,3000,3001,3122,3500,3630,3631,3764,4178,4320,4321
%N 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).
%C A268742 The sequence lists all m, in increasing order, such that floor(m/2) + floor(m/3) is a square.
%H A268742 Bruno Berselli, <a href="/A268742/b268742.txt">Table of n, a(n) for n = 0..1000</a>
%H A268742 <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 A268742 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 A268742 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 A268742 a(4*k)   = 30*k^2;
%F A268742 a(4*k+1) = 30*k^2 + 1;
%F A268742 a(4*k+2) = 30*k^2 + 12*k + 2;
%F A268742 a(4*k+3) = 30*k^2 + 48*k + 20.
%t A268742 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]
%t A268742 LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0,1,2,20,30,31,44,98,120},50] (* _Harvey P. Dale_, Dec 30 2024 *)
%o A268742 (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 A268742 (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 A268742 (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 A268742 (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 A268742 Cf. A010761.
%Y A268742 Cf. A268251: nonnegative m for which floor(m/2)*floor(m/3) is a square.
%K A268742 nonn,easy
%O A268742 0,3
%A A268742 _Bruno Berselli_, Feb 12 2016