A034198 Number of binary codes (not necessarily linear) of length n with 3 words.
0, 1, 3, 6, 10, 16, 23, 32, 43, 56, 71, 89, 109, 132, 158, 187, 219, 255, 294, 337, 384, 435, 490, 550, 614, 683, 757, 836, 920, 1010, 1105, 1206, 1313, 1426, 1545, 1671, 1803, 1942, 2088, 2241, 2401, 2569, 2744, 2927, 3118, 3317, 3524, 3740
Offset: 1
Keywords
Examples
Let t denote the trivial representation and u_1, u_2, u_3 the three nontrivial irreducible representations of C_2^2 (so the u_i are all equivalent up to automorphisms of C_2^2). Then the a(4) = 6 faithful representations of dimension 4 are:
2t+u_1+u_2; t+2u_1+u_2; t+u_1+u_2+u_3;
3u_1+u_2; 2u_1+2u_2; 2u_1+u_2+u_3.
From _John M. Campbell_, Jan 22 2016: (Start)
Letting n=8, there are a([n/2])=a(4)=6 partitions mu such that there exists a Klein four-subgroup G of S_n such that the i^th largest (nontrivial) product of 2-cycles in G consists of mu_i 2-cycles, as indicated below:
{2, 1, 1} <-> {(12)(34), (12), (34), id}
{3, 2, 1} <-> {(12)(34)(56), (34)(56), (12), id}
{2, 2, 2} <-> {(12)(34), (34)(56), (56)(12), id}
{4, 3, 1} <-> {(12)(34)(56)(78), (34)(56)(78), (12), id}
{4, 2, 2} <-> {(12)(34)(56)(78), (56)(78), (12)(34), id}
{3, 3, 2} <-> {(12)(34)(56), (34)(56)(78), (12)(78), id}
(End)
Links
- John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.
- J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
- Jan Brandts and A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
- H. Fripertinger, Isometry Classes of Codes
- H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.
- Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619-630.
- Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).
Crossrefs
Programs
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Magma
[Floor(n*(2*n^2+21*n-6)/72): n in [1..50]]; // Vincenzo Librandi, Sep 18 2016
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Maple
A034198 := n -> iquo(n*(2*n^2+21*n-6), 72): seq(A034198(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2013
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Mathematica
Table[Floor[n (2n^2+21*n-6)/72],{n,50}] (* Harvey P. Dale, Dec 25 2011 *) LinearRecurrence[ {2,0,-1,-1,0,2,-1},{0,1,3,6,10,16,23},50] (* Harvey P. Dale, Dec 25 2011 *)
Formula
a(n) = floor(n*(2*n^2 + 21*n - 6)/72).
G.f.: (-x^5 + x^3 + x^2)/((1 - x)^2*(1 - x^2)*(1 - x^3)) = 1/((1 - x)^2*(1 - x^2)*(1 - x^3)) - 1/(1 - x)^2.
a(1) = 0, a(2) = 1, a(3) = 3, a(4) = 6, a(5) = 10, a(6) = 16, a(7) = 23, and a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7) for n >= 8. [Harvey P. Dale, Dec 25 2011]
From Irena Swanson, Feb 11 2024: (Start)
The roots of the characteristic polynomial corresponding to the above recurrence are 1, 1, 1, 1, -1, -1/2 - sqrt(-3)/2 and -1/2 + sqrt(-3)/2. The corresponding closed form is:
a(n) = -25/144 - n/12 + 7n^2/24 + n^3/36 + (-1)^n/16 + (1/18 + sqrt(-3)/54)(-1/2 - sqrt(-3)/2)^n + (1/18 - sqrt(-3)/54)(-1/2 + sqrt(-3)/2)^n for n >= 1. (End)
Extensions
Additional comments from Max Alekseyev, Jul 09 2006
Additional comments from Andrew Rupinski, Jan 20 2010
Comments