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

1,3

Comments

Number of distinct triangles on vertices of n-dimensional cube.

Also, a(n) is the number of orbits of C_2^2 subgroups of C_2^n under automorphisms of C_2^n.

Also, a(n) is the number of faithful representations of C_2^2 of dimension n up to equivalence by automorphisms of (C_2^2).

Also, a([n/2]) is equal to the number of partitions mu such that there exists a C_2^2 subgroup G of S_n such that the i^th largest (nontrivial) product of 2-cycles in G consists of mu_i 2-cycles (see below example). - John M. Campbell, Jan 22 2016

Links

Table of n, a(n) for n=1..48.

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).

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)

Example

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)

Maple

A034198 := n -> iquo(n*(2*n^2+21*n-6), 72):
seq(A034198(n), n=1..100); # Wesley Ivan Hurt, Oct 29 2013

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 *)

Prog

(Magma) [Floor(n*(2*n^2+21*n-6)/72): n in [1..50]]; // Vincenzo Librandi, Sep 18 2016

Crossrefs

Cf. A034188.

Column k=2 of both A034356 and A076831 (which are the same except for column k=0).

Sequence in context: A025729 A011913 A024531 * A121776 A310085 A310086

Adjacent sequences: A034195 A034196 A034197 * A034199 A034200 A034201

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

Additional comments from Max Alekseyev, Jul 09 2006

Additional comments from Andrew Rupinski, Jan 20 2010

Status

approved