A279778 -id:A279778 - 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

A279780 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 8/5.

Original entry on oeis.org

1, -3, 5, -9, 17, -30, 52, -90, 154, -262, 446, -758, 1286, -2182, 3702, -6278, 10646, -18054, 30614, -51910, 88022, -149254, 253078, -429126, 727638, -1233798, 2092054, -3547334, 6014934, -10199046, 17293718, -29323590, 49721686, -84309126, 142956310

Offset: 0

Clark Kimberling, Dec 18 2016

If n > 4, then a(n) is even.

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

Cf. A279634, A279778, A279779, A279781, A289260.

z = 50; f[x_] := f[x] = Sum[Floor[(8/5)*(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 = 8/5.

G.f.: (1 - x) (1 - x^5)/(1 + 2 x + x^2 + 2 x^3 + 2 x^4).

A279779 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 7/5.

Original entry on oeis.org

1, -2, 0, 3, -3, 0, 4, -7, 5, 5, -16, 15, 2, -26, 39, -19, -37, 88, -73, -28, 160, -207, 61, 249, -484, 339, 258, -950, 1063, -99, -1593, 2628, -1469, -1996, 5492, -5287, -763, 9837, -14008, 5671, 14034, -31042, 25319, 11389, -59053, 73040, -16961, -92844

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,-2,-1,-2).

Cf. A279634, A279778, A279780, A279781.

z = 50; f[x_] := f[x] = Sum[Floor[(7/5)*(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 = 7/5.

G.f.: (1 - x) (1 - x^5)/(1 + x + 2 x^2 + x^3 + 2 x^4).

A279781 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 9/5.

Original entry on oeis.org

1, -3, 4, -4, 4, -3, -1, 8, -16, 24, -30, 28, -12, -20, 68, -128, 184, -208, 168, -32, -224, 592, -1008, 1344, -1408, 960, 224, -2240, 4928, -7744, 9664, -9216, 4736, 5120, -20608, 39936, -58368, 67840, -57600, 16384, 63488, -180224, 315904, -431104, 463872

Offset: 0

Clark Kimberling, Dec 18 2016

If n >= 23, then 32 divides a(n).

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

Cf. A279634, A279778, A279779, A279780.

z = 50; f[x_] := f[x] = Sum[Floor[(9/5)*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[1/f[x], {x, 0, z}], x]
LinearRecurrence[{-2,-2,-2,-2},{1,-3,4,-4,4,-3,-1},50] (* Harvey P. Dale, Jul 07 2025 *)

G.f.: 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 9/5.

G.f.: (1 - x) (1 - x^5)/(1 + 2 x + 2 x^2 + 2 x^3 + 2 x^4).

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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A279780 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 8/5.

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A279779 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 7/5.

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A279781 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 9/5.

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