

A171971


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


6



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
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OFFSET

1,3


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


REFERENCES

Mohammad K. Azarian, A Trigonometric Characterization of Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96. Solution published in Vol. 32, No. 1, Winter 1998, pp. 8485.
Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337. Solution published in Vol. 29, No. 3, Fall 1995, pp. 324325.


LINKS

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


FORMULA

a(n) = floor(n^2 * sqrt(3) / 4).
a(n)*A171974(n)/3 <= A171973(n);
A171970(n)*A004526(n) <= a(n).


PROG

(Haskell)
a171971 = floor . (/ 4) . (* sqrt 3) . fromInteger . a000290
 Reinhard Zumkeller, Dec 15 2012
(PARI) a(n)=sqrtint(3*n^4\16) \\ Charles R Greathouse IV, Apr 08 2013


CROSSREFS

Cf. A171972, A022838, A000290.
Sequence in context: A033439 A194082 A061786 * A184009 A105334 A249736
Adjacent sequences: A171968 A171969 A171970 * A171972 A171973 A171974


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Jan 20 2010


STATUS

approved



