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
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
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 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)) = 1/((1-x)^2*(1+x+x^2)).
From Paul Barry, May 19 2004: (Start)
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
MAPLE
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 *)
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
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved