A268251 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).

0, 1, 2, 3, 50, 243, 4802, 23763, 470450, 2328483, 46099202, 228167523, 4517251250, 22358088723, 442644523202, 2190864527283, 43374646022450, 214682365584963, 4250272665676802, 21036680962799043, 416483346590304050, 2061380051988721203, 40811117693184120002

Offset

0,3

Comments

Conjecture: The sequence lists all nonnegative m, in increasing order, such that floor(m/2)*floor(m/3) is a square.

This conjecture has been proved by Robert Israel (see paper in Links section).

Links

Robert Israel, Table of n, a(n) for n = 0..1000

Robert Israel, Proof of a conjecture.

Index entries for linear recurrences with constant coefficients, signature (0,99,0,-99,0,1).

Formula

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.)

a(n) = 99*a(n-2) - 99*a(n-4) + a(n-6) for n>7.

a(n) = -a(n-1) + 98*a(n-2) + 98*a(n-3) - a(n-4) - a(n-5) - 144 for n>6.

Maple

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):
S:= series(gf, x, 51):
seq(coeff(S, x, j), j=0..50); # Robert Israel, Feb 11 2016

Mathematica

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 *)

Prog

(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()
(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)))
(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);
(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)));

Crossrefs

Cf. A010762.

Cf. A268742: m for which floor(m/2) + floor(m/3) is a square.

Sequence in context: A041133 A113700 A065646 * A090508 A126716 A355648

Adjacent sequences: A268248 A268249 A268250 * A268252 A268253 A268254

Keyword

nonn,easy

Author

Bruno Berselli, Jan 29 2016

Status

approved