

A008620


Positive integers repeated three times.


31



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
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OFFSET

0,4


COMMENTS

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


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 ErrorCorrecting 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
E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's Theorem on SelfDual Codes, IEEE Trans. Information Theory, IT18 (1972), 409414.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 210 [Dead link]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 449 [Dead link]
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of selfdual codes, IEEE Trans. Inform. Theory, 18 (1972), 794805; see p. 802, col. 2, foot.
G. Nebe, E. M. Rains and N. J. A. Sloane, SelfDual 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).


FORMULA

a(n) = floor(n/3) + 1.
a(n) = A010766(n+3, 3).
G.f.: 1/((1x)*(1x^3)).
a(n) = A001840(n+1)  A001840(n).  Reinhard Zumkeller, Aug 01 2002
From Paul Barry, May 19 2004: (Start)
Convolution of A049347 and A000027.
G.f.: 1/((1x)^2*(1+x+x^2));
a(n) = Sum_{k=0..n} (k+1)*2*sqrt(3)*cos(2*Pi*(nk)/3 + Pi/6)/3. (End)
The g.f. is 1/(1V_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
a(n) = (1/9)*Sum{k=0..n}(2*(k mod 3) + ((k+1) mod 3) + 4*((k+2) mod 3]), with n>=0.  Paolo P. Lava, Nov 21 2008


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 *)


PROG

(PARI) a(n)=n\3+1
(MAGMA) [Floor(n/3)+1: n in [0..80]]; // Vincenzo Librandi, Aug 16 2011
(Haskell)
a008620 = (+ 1) . (`div` 3)
a008620_list = concatMap (replicate 3) [1..]
 Reinhard Zumkeller, Feb 19 2013, Apr 16 2012, Sep 25 2011
(Sage) def a(n) : return( dimension_modular_forms( Gamma1(3), n) ); # Michael Somos, Apr 01 2015
(MAGMA) a := func< n  Dimension( ModularForms( Gamma1(3), n))>; /* Michael Somos, Apr 01 2015 */


CROSSREFS

Cf. A008621, A002264.
Sequence in context: A086161 A002264 * A104581 A261916 A113675 A020912
Adjacent sequences: A008617 A008618 A008619 * A008621 A008622 A008623


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



