A008620 Positive integers repeated three times.

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Arises from Gleason's theorem on self-dual codes: the Molien series is 1/((1-x^8)*(1-x^24)) for the weight enumerators of doubly-even binary self-dual codes; also 1/((1-x^4)*(1-x^12)) for ternary self-dual codes.

The number of partitions of n into distinct parts where each part is either a power of two or three times a power of two.

Number of partitions of n into parts 1 or 3. - Reinhard Zumkeller, Aug 15 2011

The dimension of the space of modular forms on Gamma_1(3) of weight n>0 with a(q) the generator of weight 1 and c(q)^3 / 27 the generator of weight 3 where a(), c() are cubic AGM theta functions. - Michael Somos, Apr 01 2015

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

a(n-1) is the minimal number of circles that can be drawn through n points in general position in the plane. - Anton Zakharov, Dec 31 2016

Number of partitions of n into distinct parts from A029744.- R. J. Mathar, Mar 01 2023

Number of representations n=sum_i c_i*2^i with c_i in {0,1,3,4} [Anders]. See A120562 or A309025 for other c_i sets. - R. J. Mathar, Mar 01 2023

References

• G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 12 Exer. 7
• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
• F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Eq. (14), p. 601 and Theorem 3c, p. 602; also Problem 5 p. 620.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 1.
• E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's Theorem on Self-Dual Codes, IEEE Trans. Information Theory, IT-18 (1972), 409-414.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 210
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 449
• F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
• G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A000027, A001840, A002264, A004016, A004396, A005882, A005928, A008621, A010766, A049347.

Column 3 of A235791.

Programs

Haskell

a008620 = (+ 1) . (`div` 3)
a008620_list = concatMap (replicate 3) [1..]
-- Reinhard Zumkeller, Feb 19 2013, Apr 16 2012, Sep 25 2011

Magma

[Floor(n/3)+1: n in [0..80]]; // Vincenzo Librandi, Aug 16 2011

Magma

a := func< n | Dimension( ModularForms( Gamma1(3), n))>; /* Michael Somos, Apr 01 2015 */

Maple

A008620:=n->floor(n/3)+1; seq(A008620(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

Table[Floor[n/3] + 1, {n, 0, 90}] (* Stefan Steinerberger, Apr 02 2006 *)
Table[{n, n, n}, {n, 30}] // Flatten (* Harvey P. Dale, Jan 15 2017 *)
Ceiling[Range[20]/3] (* Eric W. Weisstein, Aug 12 2023 *)
Table[Ceiling[n/3], {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(1 + n - Cos[2 n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n, -1/2] + 1)/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 1, 2}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[1/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)

PARI

a(n)=n\3+1

Sage

def a(n) : return( dimension_modular_forms( Gamma1(3), n) ); # Michael Somos, Apr 01 2015

Formula

a(n) = floor(n/3) + 1.

a(n) = A010766(n+3, 3).

G.f.: 1/((1-x)*(1-x^3)) = 1/((1-x)^2*(1+x+x^2)).

a(n) = A001840(n+1) - A001840(n). - Reinhard Zumkeller, Aug 01 2002

From Paul Barry, May 19 2004: (Start)

Convolution of A049347 and A000027.

a(n) = Sum_{k=0..n} (k+1)*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3. (End)

The g.f. is 1/(1-V_trefoil(x)), where V_trefoil is the Jones polynomial of the trefoil knot. - Paul Barry, Oct 08 2004

a(2n) = A004396(n+1). - Philippe Deléham, Dec 14 2006

a(n) = ceiling(n/3), n>=1. - Mohammad K. Azarian, May 22 2007

E.g.f.: exp(x)*(2 + x)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022