A171974 -id:A171974 - cp's OEIS frontend

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

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.

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).

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).

A325731 First term of n-th difference sequence of (floor(k*r)), r = sqrt(2/3), k >= 0.

Original entry on oeis.org

0, 1, -1, 1, -1, 0, 5, -20, 55, -125, 250, -450, 725, -1000, 1000, 1, -3642, 13278, -35344, 81720, -174098, 352716, -694904, 1355322, -2652200, 5245956, -10491911, 21097440, -42262518, 83482620, -161013975, 300338820, -536165400, 903201960, -1401448500

Offset: 1

Clark Kimberling, May 20 2019

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

Cf. A325664. Inverse binomial transform of A171974.

Table[First[Differences[Table[Floor[Sqrt[2/3]*n], {n, 0, 50}], n]], {n, 1, 50}]

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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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A171973 Integer part of the volume 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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A325731 First term of n-th difference sequence of (floor(k*r)), r = sqrt(2/3), k >= 0.

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