A171972 -id:A171972 - 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).

A171973 Integer part of the volume of a regular tetrahedron with edge length n.

Original entry on oeis.org

0, 0, 3, 7, 14, 25, 40, 60, 85, 117, 156, 203, 258, 323, 397, 482, 579, 687, 808, 942, 1091, 1254, 1433, 1629, 1841, 2071, 2319, 2587, 2874, 3181, 3510, 3861, 4235, 4632, 5052, 5498, 5969, 6466, 6990, 7542, 8122, 8731, 9369, 10039, 10739, 11471, 12235

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

Lim{n->oo} a(n)/A000292(n) = sqrt(2)/2;

floor(A171971(n)*A171974(n)/3) <= a(n).

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

Cf. A001951, A171972, A171975, A000578.

a171973 = floor . (/ 12) . (* sqrt 2) . fromInteger . a000578
-- Reinhard Zumkeller, Dec 15 2012

a(n) = floor(n^3 * sqrt(2) / 12).

A171970 Integer part of the height of an equilateral triangle with side length n.

Original entry on oeis.org

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, 63

Offset: 1

Reinhard Zumkeller, Jan 20 2010

Eric Weisstein's World of Mathematics, Equilateral Triangle
Wikipedia, Equilateral triangle

Beatty sequence of A010527.

Cf. A022838, A004526, A005843, A171971, A171972.

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

a(n) = floor(n*sqrt(3)/2).
a(n) = floor(A022838(n)/2).
a(n)*A004526(n) <= A171971(n)
a(n)*A005843(n) <= A171972(n).

A171974 Integer part of the height of a regular tetrahedron with edge length n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

-3 <= 4*A171975(n) - 3*a(n) < 3;
a(n)*A171975(n) <= A007590(n);
floor(a(n)*A171971(n)/3) <= A171973(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron
Index entries for sequences related to Beatty sequences

Cf. A171972, A022840. Beatty sequence of A157697.

Cf. A007590, A171971, A171973, A171975.

a171974 = floor . (/ 3) . (* sqrt 6) . fromInteger
-- Reinhard Zumkeller, Dec 15 2012

a(n) = floor(n*sqrt(6)/3).

A171975 Integer part of the circumsphere radius of a regular tetrahedron with edge length n.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 41, 41, 42, 42, 43, 44, 44, 45, 45

Offset: 1

Reinhard Zumkeller, Jan 20 2010

nonn

-3 <= 4*a(n) - 3*A171974(n) < 3;

a(n)*A171974(n) <= A007590(n).

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

Cf. A171973, A171972, A022840. Beatty sequence of A187110.

a171975 = floor . (/ 4) . (* sqrt 6) . fromInteger
-- Reinhard Zumkeller, Dec 15 2012

a(n) = floor(n*sqrt(6)/4).

A293410 Least integer k such that k/n^2 > sqrt(3).

Original entry on oeis.org

0, 2, 7, 16, 28, 44, 63, 85, 111, 141, 174, 210, 250, 293, 340, 390, 444, 501, 562, 626, 693, 764, 839, 917, 998, 1083, 1171, 1263, 1358, 1457, 1559, 1665, 1774, 1887, 2003, 2122, 2245, 2372, 2502, 2635, 2772, 2912, 3056, 3203, 3354, 3508, 3666, 3827, 3991

Offset: 0

Clark Kimberling, Oct 11 2017

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

Cf. A002194, A171972, A070169.

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 *)

from math import isqrt
def A293410(n): return 1+isqrt(3*n**4-1) if n else 0 # Chai Wah Wu, Jul 31 2022

a(n) = ceiling(r*n^2), where r = sqrt(3).

a(n) = A171972(n) + 1 for n > 0.

cp's OEIS Frontend

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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A171973 Integer part of the volume of a regular tetrahedron with edge length n.

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A171970 Integer part of the height of an equilateral triangle with side length n.

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A171974 Integer part of the height of a regular tetrahedron with edge length n.

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A171975 Integer part of the circumsphere radius of a regular tetrahedron with edge length n.

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A293410 Least integer k such that k/n^2 > sqrt(3).

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Python

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