A279678 -id:A279678 - cp's OEIS frontend

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

1, -3, 5, -9, 18, -36, 72, -144, 288, -576, 1152, -2304, 4608, -9216, 18432, -36864, 73728, -147456, 294912, -589824, 1179648, -2359296, 4718592, -9437184, 18874368, -37748736, 75497472, -150994944, 301989888, -603979776, 1207959552, -2415919104, 4831838208

Offset: 0

Clark Kimberling, Dec 18 2016

After first 3 terms, agrees with A005010 except for signs; in particular 9 divides a(n) for n >= 3.
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.
3/2    A279634     yes
4/3    A279675     no
5/3    A279676     no
5/4    A279677     yes
7/4    A279678     yes
6/5    A279778     no
7/5    A279779     no
8/5    A279780     yes
9/5    A279781     no

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A005010.

z = 50; f[x_] := f[x] = Sum[Floor[(3/2)*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[1/f[x], {x, 0, z}], x]
LinearRecurrence[{-2},{1,-3,5,-9},40] (* Harvey P. Dale, Jul 28 2023 *)

G.f.: 1/(1 + 3x + 4x^2 + 6x^3 + ...).
G.f.: (1 - x) (1 - x^2)/(1 + 2x).
E.g.f.: - (1/8) - (3/4)*x + (1/4)*x^2 + (9/8)*exp(-2*x). - Alejandro J. Becerra Jr., Feb 16 2021

A279677 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 5/4.

Original entry on oeis.org

1, -2, 1, -1, 3, -3, 2, -5, 9, -8, 9, -19, 26, -25, 37, -64, 77, -87, 138, -205, 241, -312, 481, -651, 794, -1105, 1613, -2096, 2693, -3823, 5322, -6885, 9209, -12968, 17529, -22979, 31386, -43465, 58037, -77344, 106237, -144967, 193418, -260925, 357441

Offset: 0

Clark Kimberling, Dec 18 2016

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-2).

Cf. A279634, A279678.

z = 50; f[x_] := f[x] = Sum[Floor[(5/4)*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[1/f[x], {x, 0, z}], x]

G.f.: 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 5/4.

G.f.: (1 - x) (1 - x^4)/(1 + x + x^2 + 2 x^3).

cp's OEIS Frontend

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

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