A054385 -id:A054385 - cp's OEIS frontend

A014246 a(n) = (n-th term of Beatty sequence for e) - (n-th term of Beatty sequence for e/(e-1)).

1, 2, 4, 4, 6, 7, 8, 9, 10, 12, 12, 14, 15, 16, 17, 18, 20, 20, 21, 23, 24, 25, 26, 28, 28, 29, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 45, 47, 48, 48, 50, 51, 53, 53, 55, 56, 56, 58, 59, 61, 61, 62, 64, 64, 66, 67, 69, 69, 70, 72, 72, 74, 75, 77, 77, 78, 80, 80, 82

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Beatty sequences

Cf. A022843, A054385.

E:=Exp(1); [Floor(n*E) - Floor(n*E/(E-1)): n in [1..75]]; // G. C. Greubel, Jun 19 2019

Table[Floor[n*E] - Floor[n*E/(E-1)], {n,1,75}] (* G. C. Greubel, Jun 19 2019 *)

my(e=exp(1)); vector(75, n, (n*e)\1 - (n*e)\(e-1)) \\ G. C. Greubel, Jun 19 2019

[floor(n*e) - floor(n*e/(e-1)) for n in (1..75)] # G. C. Greubel, Jun 19 2019

a(n) = floor(e * n) - floor(e/(e-1) * n). - Franklin T. Adams-Watters, Nov 02 2006

Corrected and extended by Franklin T. Adams-Watters, Nov 02 2006

A279632 Coefficients in the expansion of ([s] + [2s]x + [3s]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1, s = r/(1-r).

Original entry on oeis.org

2, -2, 3, -2, -2, 8, -14, 17, -12, -5, 34, -68, 91, -80, 11, 126, -308, 467, -488, 235, 382, -1316, 2291, -2760, 1995, 638, -5220, 10738, -14725, 13447, -3007, -18467, 47914, -74806, 80821, -43890, -51936, 201548, -363193, 450980, -347117, -55972, 782359

Offset: 0

Clark Kimberling, Dec 18 2016

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000210, A054385.

z = 100;
r = E - 1; f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];
s = r/(r - 1); g[x_] := g[x] = Sum[Floor[s*(k + 1)] x^k, {k, 0, z}]
CoefficientList[Series[g[x]/f[x], {x, 0, z}], x]

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

A022806 a(n) = B(n) + c(n) where B(n) is Beatty sequence [ n*e ] and c is the complement of B.

Original entry on oeis.org

3, 8, 12, 16, 20, 25, 30, 33, 38, 42, 46, 50, 55, 60, 63, 68, 72, 76, 81, 85, 90, 93, 98, 102, 106, 111, 115, 120, 123, 128, 133, 136, 141, 145, 150, 153, 158, 163, 167, 171, 175, 180, 184, 188, 193, 197, 201, 205, 210, 214, 218, 223, 227, 231, 236

Offset: 1

Clark Kimberling

nonn

Index entries for sequences related to Beatty sequences

a(n) = A022843(n + 1) + A054385(n). - Sean A. Irvine, May 21 2019

A054645 Triangle T(n,k) of asymmetric n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

Original entry on oeis.org

0, 0, 2, 6, 10, 10, 6, 2, 0, 0, 0, 1, 10, 63, 207, 525, 954, 1395, 1550, 1395, 954, 525, 207, 63, 10, 1, 0, 0, 1, 28, 258, 1503, 6475, 21810, 59540, 134333, 254178, 407040, 555356, 648054, 648054, 555356, 407040, 254178, 134333, 59540, 21810, 6475

Offset: 3

Vladeta Jovovic, May 15 2000

Row sums give A054385.

			The batch [0,0,2,6,10,10,6,2,0,0] gives the numbers of asymmetric 3 X 3 binary matrices with k=0..9 ones under action of dihedral group of the square D_4.
There are 6 nonequivalent asymmetric 3 X 3 binary matrices with 3 ones under action of D_4:
[0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 1]
[0 0 1] [0 0 1] [0 1 0] [0 1 1] [1 0 1] [0 0 0]
[1 0 1] [1 1 0] [0 1 1] [1 0 0] [0 0 1] [1 1 0].

Cf. A054252.

cp's OEIS Frontend

A014246 a(n) = (n-th term of Beatty sequence for e) - (n-th term of Beatty sequence for e/(e-1)).

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A279632 Coefficients in the expansion of ([s] + [2s]x + [3s]x^2 + ...)/([r] + [2r]x + [3r]x^2 + ...); [ ] = floor, r = e - 1, s = r/(1-r).

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A022806 a(n) = B(n) + c(n) where B(n) is Beatty sequence [ n*e ] and c is the complement of B.

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Formula

A054645 Triangle T(n,k) of asymmetric n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

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