A171971 -id:A171971 - cp's OEIS frontend

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

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).

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).

A255604 Table read by antidiagonals, T(n,k) is the integer part of the area of a regular k-gon with side length n.

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 6, 9, 6, 2, 10, 16, 15, 10, 3, 15, 25, 27, 23, 14, 4, 21, 36, 43, 41, 32, 19, 6, 27, 49, 61, 64, 58, 43, 24, 7, 35, 64, 84, 93, 90, 77, 55, 30, 9, 43, 81, 110, 127, 130, 120, 98, 69, 37, 11, 52, 100, 139, 166, 178, 173, 154, 123, 84, 44, 13, 62, 121, 172

Offset: 1

Kival Ngaokrajang, Feb 27 2015

See illustration in the links.

			See table in the links.

Kival Ngaokrajang, Illustration of T(n,k), n = 1..4, k = 3..6, Example of table

Row 1: A011865.

Columns 1,2,3,4,10: A171971, A000290, A255605, A255606, A172526.

t[n_, k_] := Floor[k*n^2/(4 Tan[Pi/k])]; Table[t[n - k + 1, k], {n, 3, 14}, {k, 3 , n}] // Flatten
(* to view table: Table[t[n, k], {k, 3, 6}, {n, 6}] // TableForm *) (* Robert G. Wilson v, Feb 28 2015 *)

{for(i=1,20,for(n=3,i-1,a=floor(n*(i-n)^2/(4*tan(Pi/n)));print1(a,", ")))}

T(n,k) = floor(k*n^2/(4*tan(Pi/k))), n >=1, k >=3.

A308358 Beatty sequence for sqrt(3)/4.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 29, 30, 30, 31, 31, 32, 32, 32, 33, 33, 34, 34

Offset: 0

R. J. Mathar, May 22 2019

Differs from A057357 first at n=37.

Index to sequences related to Beatty sequences

Cf. A120011.

Floor[Sqrt[3] Range[0, 100]/4] (* Wesley Ivan Hurt, Dec 26 2023 *)

a(n) = floor(n*A120011).

A171971(n) = a(n^2).

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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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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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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A255604 Table read by antidiagonals, T(n,k) is the integer part of the area of a regular k-gon with side length n.

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A308358 Beatty sequence for sqrt(3)/4.

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

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