A279780 -id:A279780 - 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

A279778 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 6/5.

Original entry on oeis.org

1, -2, 1, 0, -1, 3, -3, 1, 1, -5, 9, -7, 1, 7, -19, 25, -15, -5, 33, -63, 65, -25, -43, 129, -191, 155, -7, -215, 449, -537, 317, 201, -879, 1435, -1391, 433, 1281, -3193, 4261, -3215, -415, 5755, -10647, 11737, -6015, -6585, 22157, -33031, 29489, -5445

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,-1,-2).

Cf. A279634, A279779, A279780, A279781.

z = 50; f[x_] := f[x] = Sum[Floor[(6/5)*(k + 1)] x^k, {k, 0, z}]; f[x]
CoefficientList[Series[1/f[x], {x, 0, z}], x]
LinearRecurrence[{-1,-1,-1,-2},{1,-2,0,-1,3,-3},50] (* Harvey P. Dale, Mar 11 2024 *)

G.f.: 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 6/5.

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

A289260 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, 242399686, 411017942

Offset: 0

Clark Kimberling, Jul 14 2017

Conjecture: the sequence is strictly increasing.

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

Cf. A078140 (includes guide to related sequences), A289265.

Cf. A279780.

r = 8/5;
u = 1000; (* # initial terms from given series *)
v = 100;   (* # coefficients in reciprocal series *)
CoefficientList[Series[1/Sum[Floor[r*(k + 1)] (-x)^k, {k, 0, u}], {x, 0, v}], x]
LinearRecurrence[{2,-1,2,-2},{1,3,5,9,17,30,52},40] (* Harvey P. Dale, Oct 13 2023 *)

Vec((1 + x)^2*(1 - x + x^2 - x^3 + x^4) / ((1 - x)*(1 - x - 2*x^3)) + O(x^50)) \\ Colin Barker, Jul 20 2017

G.f.: 1/(Sum_{k>=0} [(k+1)*r](-x)^k), where r = 8/5 and [ ] = floor.
From Colin Barker, Jul 14 2017: (Start)
G.f.: (1 + x)^2*(1 - x + x^2 - x^3 + x^4) / ((1 - x)*(1 - x - 2*x^3)).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-4) for n>3.
(End)
a(n) = abs(A279780(n)). - Alois P. Heinz, Jul 15 2017

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

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A279778 Coefficients in the expansion of 1/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = 6/5.

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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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A289260 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=8/5.

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