A171971 - OEIS

%I #27 Feb 16 2025 08:33:11

%S 0,1,3,6,10,15,21,27,35,43,52,62,73,84,97,110,125,140,156,173,190,209,

%T 229,249,270,292,315,339,364,389,416,443,471,500,530,561,592,625,658,

%U 692,727,763,800,838,876,916,956,997,1039,1082,1126,1170,1216,1262,1309

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

%C 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

%H Reinhard Zumkeller, <a href="/A171971/b171971.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EquilateralTriangle.html">Equilateral Triangle</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Equilateral_triangle">Equilateral triangle</a>

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

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

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

%t Table[Floor[(n^2 Sqrt[3])/4],{n,60}] (* _Harvey P. Dale_, Apr 13 2022 *)

%o (Haskell)

%o a171971 = floor . (/ 4) . (* sqrt 3) . fromInteger . a000290

%o -- _Reinhard Zumkeller_, Dec 15 2012

%o (PARI) a(n)=sqrtint(3*n^4\16) \\ _Charles R Greathouse IV_, Apr 08 2013

%Y Cf. A171972, A022838, A000290.

%K nonn,changed

%O 1,3

%A _Reinhard Zumkeller_, Jan 20 2010