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A008620 Positive integers repeated three times. 28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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]

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.

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.

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

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 to sequences with 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/((1-x)*(1-x^3)).

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

Convolution of A049347 and A000027. G.f.: 1/((1-x)^2*(1+x+x^2)); a(n)=sum{k=0..n, (k+1)*2*sqrt(3)*cos(2*Pi*(n-k)/3+Pi/6)/3}. - Paul Barry, May 19 2004

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

a(n) = (1/9)*Sum{k=0..n}{-2*(k mod 3)+[(k+1) mod 3]+4*[(k+2) mod 3]}, with n>=0. [From 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

CROSSREFS

Cf. A008621, A002264.

Sequence in context: A086161 A002264 * A104581 A113675 A020912 A194990

Adjacent sequences:  A008617 A008618 A008619 * A008621 A008622 A008623

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 26 01:20 EDT 2014. Contains 248566 sequences.