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%I #40 Sep 08 2022 08:46:15
%S 0,1,2,3,50,243,4802,23763,470450,2328483,46099202,228167523,
%T 4517251250,22358088723,442644523202,2190864527283,43374646022450,
%U 214682365584963,4250272665676802,21036680962799043,416483346590304050,2061380051988721203,40811117693184120002
%N Expansion of x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6).
%C Conjecture: The sequence lists all nonnegative m, in increasing order, such that floor(m/2)*floor(m/3) is a square.
%C This conjecture has been proved by _Robert Israel_ (see paper in Links section).
%H Robert Israel, <a href="/A268251/b268251.txt">Table of n, a(n) for n = 0..1000</a>
%H Robert Israel, <a href="/A268251/a268251.pdf">Proof of a conjecture</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,99,0,-99,0,1).
%F G.f.: x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/((1 - x)*(1 + x)*(1 - 10*x + x^2)*(1 + 10*x + x^2)). (For e.g.f see Israel's paper.)
%F a(n) = 99*a(n-2) - 99*a(n-4) + a(n-6) for n>7.
%F a(n) = -a(n-1) + 98*a(n-2) + 98*a(n-3) - a(n-4) - a(n-5) - 144 for n>6.
%p gf:= x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6):
%p S:= series(gf,x,51):
%p seq(coeff(S,x,j),j=0..50); # _Robert Israel_, Feb 11 2016
%t CoefficientList[x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6) + O[x]^30, x] (* _Jean-François Alcover_, Feb 12 2016 *)
%o (Sage) gf = x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6); taylor(gf, x, 0, 30).list()
%o (PARI) concat(0, Vec((1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6) + O(x^30)))
%o (Maxima) makelist(coeff(taylor(x*(1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6), x, 0, n), x, n), n, 0, 30);
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1 + 2*x - 96*x^2 - 148*x^3 + 45*x^4 + 50*x^5 + 2*x^6)/(1 - 99*x^2 + 99*x^4 - x^6)));
%Y Cf. A010762.
%Y Cf. A268742: m for which floor(m/2) + floor(m/3) is a square.
%K nonn,easy
%O 0,3
%A _Bruno Berselli_, Jan 29 2016
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