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

Author

Mitch Harris

Keywords

• nonn
• easy

Comments

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.

References

• 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.

Links

• 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.

Crossrefs

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.

Programs

Magma

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

Mathematica

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

PARI

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

Formula

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