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A279634 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 3/2.

%I A279634 #23 Jul 28 2023 15:10:40
%S A279634 1,-3,5,-9,18,-36,72,-144,288,-576,1152,-2304,4608,-9216,18432,-36864,
%T A279634 73728,-147456,294912,-589824,1179648,-2359296,4718592,-9437184,
%U A279634 18874368,-37748736,75497472,-150994944,301989888,-603979776,1207959552,-2415919104,4831838208
%N A279634 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 3/2.
%C A279634 After first 3 terms, agrees with A005010 except for signs; in particular 9 divides a(n) for n >= 3.
%C A279634 Suppose r = c/d is a rational number and (a(n)) is the coefficient series for 1/([r] + [2r]x + [3r]x^2 + ...). Let (s(k)) be the increasing sequence of indices n(k) for which a(n(k)) > = 0. In the table below, "yes" indicates that a check of the first 1000 terms indicates that (n(k)) is (eventually) periodic. Column 1 gives selected values of r, and column 2 gives the corresponding coefficient series.
%C A279634 3/2    A279634     yes
%C A279634 4/3    A279675     no
%C A279634 5/3    A279676     no
%C A279634 5/4    A279677     yes
%C A279634 7/4    A279678     yes
%C A279634 6/5    A279778     no
%C A279634 7/5    A279779     no
%C A279634 8/5    A279780     yes
%C A279634 9/5    A279781     no
%H A279634 Clark Kimberling, <a href="/A279634/b279634.txt">Table of n, a(n) for n = 0..1000</a>
%H A279634 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-2).
%F A279634 G.f.: 1/(1 + 3x + 4x^2 + 6x^3 + ...).
%F A279634 G.f.: (1 - x) (1 - x^2)/(1 + 2x).
%F A279634 E.g.f.: - (1/8) - (3/4)*x + (1/4)*x^2 + (9/8)*exp(-2*x). - _Alejandro J. Becerra Jr._, Feb 16 2021
%t A279634 z = 50; f[x_] := f[x] = Sum[Floor[(3/2)*(k + 1)] x^k, {k, 0, z}]; f[x]
%t A279634 CoefficientList[Series[1/f[x], {x, 0, z}], x]
%t A279634 LinearRecurrence[{-2},{1,-3,5,-9},40] (* _Harvey P. Dale_, Jul 28 2023 *)
%Y A279634 Cf. A005010.
%K A279634 sign,easy
%O A279634 0,2
%A A279634 _Clark Kimberling_, Dec 18 2016