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 A008620 Positive integers repeated three times. 35
 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 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 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 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)). 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/((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. (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 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 *) Table[{n, n, n}, {n, 30}]//Flatten (* Harvey P. Dale, Jan 15 2017 *) 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. Column 3 of A235791. Sequence in context: A296357 A086161 A002264 * A104581 A261916 A113675 Adjacent sequences:  A008617 A008618 A008619 * A008621 A008622 A008623 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified June 24 05:58 EDT 2021. Contains 345416 sequences. (Running on oeis4.)