A308358 -id:A308358 - cp's OEIS frontend

A057357 a(n) = floor(3*n/7).

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, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

This sequence relates to 3/7 = 0.42857142... (essentially given by A020806). It differs from the Beatty sequence A308358 for sqrt(3)/4 = 0.43301270... = A120011.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).
Index to sequences related to Beatty sequences.

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A020806, A193535, A308358.

[Floor(3*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[(3*Range[0,80])/7] (* Harvey P. Dale, Aug 22 2013 *)

a(n)=3*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x^2+x^4)*x^3/((1-x)*(1-x^7)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
for all m>=0 a(7m)=0 mod 3; a(7m+1)=0 mod 3; a(7m+2)= 0 mod 3; a(7m+3) = 1 mod 3; a(5m+4) = 1 mod 3; a(7m+5) = 2 mod 3; a(7m+6) = 2 mod 3 - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2)/3 (A193535). - Amiram Eldar, Sep 30 2022

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A057357 a(n) = floor(3*n/7).

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