A057353 a(n) = floor(3n/4).

0, 0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53, 54

Offset

0,4

Comments

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

For n >= 2, a(n) is the number of different integers that can be written as floor(k^2/n) for k = 1, 2, 3, ..., n-1. Generalization of the 1st problem proposed during the 15th Balkan Mathematical Olympiad in 1998 where the question was asked for n = 1998 with a(1998) = 1498. - Bernard Schott, Apr 22 2022

For n > 1, a(n) is also the Hadwiger number of the (n+1)-cycle complement graph (up to at least n = 16). - Eric W. Weisstein, Mar 10 2025

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Balkan Mathematical Olympiad, Problem 1, 15th Balkan Mathematical Olympiad 1998.

Eric Weisstein's World of Mathematics, Cycle Complement Graph.

Eric Weisstein's World of Mathematics, Hadwiger Number.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Index entries for sequences related to Beatty sequences.

Index to sequences related to Olympiads.

Formula

G.f.: (1+x+x^2)*x^2/((1-x)*(1-x^4)). - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002

For all m>=0 a(4m)=0 mod 3; a(4m+1)=0 mod 3; a(4m+2)= 1 mod 3; a(4m+3) = 2 mod 3

a(n) = A002378(n) - A173562(n). - Reinhard Zumkeller, Feb 21 2010

a(n+1) = A140201(n) - A002265(n+1). - Reinhard Zumkeller, Jan 26 2011

a(n) = n-1 - A002265(n-1) = ( A007310(n) + A057077(n+1) )/4 for n>0. a(n) = a(n-1)+a(n-4)-a(n-5) for n>4. - Bruno Berselli, Jan 28 2011

a(n) = 1/8*(6*n + 2*cos((Pi*n)/2) + cos(Pi*n) - 2*sin((Pi*n)/2) - 3). - Ilya Gutkovskiy, Sep 18 2015

a(4n) = a(4n+1). - Altug Alkan, Sep 26 2015

Sum_{n>=2} (-1)^n/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Sep 29 2022

Mathematica

Table[Floor[3 n/4], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Floor[3 Range[0, 20]/4] (* Eric W. Weisstein, Mar 10 2025 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 2, 3, 3}, {0, 20}] (* Eric W. Weisstein, Mar 10 2025 *)
CoefficientList[Series[x^2 (1 + x + x^2)/(1 - x - x^4 + x^5), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 10 2025 *)

Prog

(Magma) [Floor(3*n/4): n in [0..90]]; // Vincenzo Librandi, Feb 12 2012
(PARI) a(n)=3*n\4 \\ Charles R Greathouse IV, Sep 02 2015

Crossrefs

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

Cf. A002378, A007310, A057077, A073010, A173562, A182210.

Sequence in context: A083544 A377982 A367328 * A076539 A074184 A187329

Adjacent sequences: A057350 A057351 A057352 * A057354 A057355 A057356

Keyword

nonn,easy,changed

Author

Mitch Harris

Status

approved