A171970 -id:A171970 - cp's OEIS frontend

A022838 Beatty sequence for sqrt(3); complement of A054406.

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112

Offset: 1

Clark Kimberling

nonn

0 <= A144077(n) - a(n) <= 1. - Reinhard Zumkeller, Sep 09 2008
From Reinhard Zumkeller, Jan 20 2010: (Start)
A080757(n) = a(n+1) - a(n).
A171970(n) = floor(a(n)/2).
A171972(n) = a(A000290(n)). (End)
Numbers k>0 such that A194979(k+1) = A194979(k) + 1. - Clark Kimberling, Dec 02 2014
Powers of 2 (i.e, 1, 8, 32, 64, 128, 256, 512, 4096, 8192,...) appear at n=1, 5, 19, 37, 74, 148, 296, 2365, 4730, 18919, 75675, 151349, 302698, 605396, ... related to A293328. - R. J. Mathar, Jan 17 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A080757 (first differences), A194106 (partial sums), A194028 (even bisection), A184796 (prime terms).

Cf. A026255, A054406 (complement).

a022838 = floor . (* sqrt 3) . fromIntegral
-- Reinhard Zumkeller, Sep 14 2014

[Floor(n*Sqrt(3)): n in [1..60]]; // G. C. Greubel, Sep 28 2018

A022838 := proc(n)
    floor(n*sqrt(3)) ;
end proc: # R. J. Mathar, Mar 25 2013

Table[Floor[n 3^(1/2)] , {n, 1, 65}] (* Geoffrey Critzer, Jan 11 2015 *)

vector(60, n, floor(n*sqrt(3))) \\ G. C. Greubel, Sep 28 2018

a(n)=sqrtint(3*n^2) \\ Charles R Greathouse IV, Nov 01 2021

from math import isqrt
def A022838(n): return isqrt(3*n*n) # Chai Wah Wu, Aug 06 2022

a(n) = floor(n*sqrt(3)). - Reinhard Zumkeller, Jan 20 2010

a(n) = 2 * floor(n * (sqrt(3) - 1)) + floor(n * (2 - sqrt(3))) + 1. - Miko Labalan, Dec 03 2016

A171971 Integer part of the area of an equilateral triangle with side length n.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 27, 35, 43, 52, 62, 73, 84, 97, 110, 125, 140, 156, 173, 190, 209, 229, 249, 270, 292, 315, 339, 364, 389, 416, 443, 471, 500, 530, 561, 592, 625, 658, 692, 727, 763, 800, 838, 876, 916, 956, 997, 1039, 1082, 1126, 1170, 1216, 1262, 1309

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

The Beatty sequence of sqrt(3)/4 starts 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7,... for n>=1. This sequence here subsamples the Beatty sequence at the positions of the squares. - R. J. Mathar, Dec 02 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Equilateral Triangle
Wikipedia, Equilateral triangle

Cf. A171972, A022838, A000290.

a171971 = floor . (/ 4) . (* sqrt 3) . fromInteger . a000290
-- Reinhard Zumkeller, Dec 15 2012

Table[Floor[(n^2 Sqrt[3])/4],{n,60}] (* Harvey P. Dale, Apr 13 2022 *)

a(n)=sqrtint(3*n^4\16) \\ Charles R Greathouse IV, Apr 08 2013

a(n) = floor(n^2 * sqrt(3) / 4) = A308358(n^2).
a(n)*A171974(n)/3 <= A171973(n);
A171970(n)*A004526(n) <= a(n).

A171972 Greatest integer k such that k/n^2 < sqrt(3).

Original entry on oeis.org

0, 1, 6, 15, 27, 43, 62, 84, 110, 140, 173, 209, 249, 292, 339, 389, 443, 500, 561, 625, 692, 763, 838, 916, 997, 1082, 1170, 1262, 1357, 1456, 1558, 1664, 1773, 1886, 2002, 2121, 2244, 2371, 2501, 2634, 2771, 2911, 3055, 3202, 3353, 3507, 3665, 3826, 3990, 4158

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

Integer part of the surface area of a regular tetrahedron with edge length n.
A171970(n)*A005843(n) <= a(n);
a(n) <= 4*A171971(n); 0 <= a(n) - 4*A171971(n) < 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron

Cf. A002194, A070169, A171974, A171973, A171975, A000290, A293410.

a171972 = floor . (* sqrt 3) . fromInteger . a000290
-- Reinhard Zumkeller, Dec 15 2012

z = 120; r = Sqrt[3];
Table[Floor[r*n^2], {n, 0, z}]; (* A171972 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293410 *)
Table[Round[r*n^2], {n, 0, z}]; (* A070169. -  Clark Kimberling, Oct 11 2017 *)

a(n) = floor(n^2 * sqrt(3)).
a(n) = A022838(n^2);
a(n) = A293410(n) - 1 for n > 0.

A172475 a(n) = floor(n*sqrt(3)/2).

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 58, 59, 60, 61, 62

Offset: 0

Vincenzo Librandi, Feb 04 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

[Floor(n*Sqrt(3)/2): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

With[{c = Sqrt[3]/2}, Floor[c #] &/@ Range[0, 90]]  (* Harvey P. Dale, Feb 07 2011 *)

a(n) = A171970(n). [R. J. Mathar, Feb 05 2010]

A325732 First term of n-th difference sequence of (floor(k*r)), r = sqrt(3/4), k >= 0.

Original entry on oeis.org

0, 1, -1, 1, -1, 1, -1, 0, 7, -35, 119, -329, 791, -1715, 3430, -6419, 11319, -18767, 28763, -38759, 38759, 1, -149228, 572057, -1615429, 3979001, -9014851, 19251001, -39309301, 77558760, -149239771, 282712561, -532577025, 1008032953, -1934671809, 3787949521

Offset: 1

Clark Kimberling, May 20 2019

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A325664. Inverse binomial Transform of A171970 and A172475.

Cf. A010527 (sqrt(3/4)).

Table[First[Differences[Table[Floor[Sqrt[3/4]*n], {n, 0, 50}], n]], {n, 1, 50}]

A194082 Sum{floor(sqrt(3)*k/2) : 1<=k<=n}.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 208, 227, 247, 268, 290, 313, 337, 362, 387, 413, 440, 468, 497, 527, 558, 590, 622, 655, 689, 724, 760, 797, 835, 873, 912, 952, 993, 1035, 1078, 1122, 1167, 1212

Offset: 1

Clark Kimberling, Aug 17 2011

nonn

Partial sums of A171970.
Comment from R. J. Mathar, Dec 02 2012 (Start):
a(n-1) is the number of unit squares regularly packed into the isosceles triangle of edge length n.
The triangle may be aligned with the Cartesian axes by putting its bottom edge on the horizontal axis, so its vertices are at (x,y) = (0,0), (n,0) and (n/2,sqrt(3)*n/2), see A010527.
The area inside the triangle is sqrt(3)*n^2/4 = A120011*n^2. There is an obvious upper limit of floor(sqrt(3)*n^2/4) = A171971(n) to the count of non-overlapping unit squares inside this triangle.
Regular packing: We place the first row of unit squares so they touch the bottom edge of the triangle. Their number is limited by the length of the horizontal section of the line y=1 inside the triangle, n-2*y/sqrt(3), which touches all of these first-row squares at their top.
The number of unit squares in the next row, between y=1 and y=2, is limited by the length of the horizontal section of the line y=2 inside the triangle, n-2*y/sqrt(3). Continuing, in row y=1, 2, ... we insert floor(n-2*y/sqrt(3)) unit squares, all with the same orientation.
The total number of squares is sum_{ y=1, 2, ..., floor(n*sqrt(3)/2) } floor( n-2*y/sqrt(3) ), and resummation yields, up to an index shift, this sequence here.
(End)

Cf. A171970.

r = Sqrt[3]/2;
c[k_] := Sum[Floor[j*r], {j, 1, k}];
Table[c[k], {k, 1, 90}]

a(n)=sum(k=1,n,sqrtint(3*k^2\4)) \\ Charles R Greathouse IV, Jan 06 2013

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A022838 Beatty sequence for sqrt(3); complement of A054406.

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A171971 Integer part of the area of an equilateral triangle with side length n.

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A171972 Greatest integer k such that k/n^2 < sqrt(3).

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A172475 a(n) = floor(n*sqrt(3)/2).

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A325732 First term of n-th difference sequence of (floor(k*r)), r = sqrt(3/4), k >= 0.

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A194082 Sum{floor(sqrt(3)*k/2) : 1<=k<=n}.

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