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

Author

Reinhard Zumkeller, Jan 20 2010

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A171972, A022838, A000290.

Programs

Haskell

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

Mathematica

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

PARI

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

Formula

a(n) = floor(n^2 * sqrt(3) / 4) = A308358(n^2).

a(n)*A171974(n)/3 <= A171973(n);

A171970(n)*A004526(n) <= a(n).