

A034198


Number of binary codes (not necessarily linear) of length n with 3 words.


7



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
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OFFSET

1,3


COMMENTS

Number of distinct triangles on vertices of ndimensional 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 2cycles in G consists of mu_i 2cycles (see below example).  John M. Campbell, Jan 22 2016


LINKS

Table of n, a(n) for n=1..48.
Campbell, John, A class of symmetric differenceclosed 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 ncube, 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, A. Cihangir, Enumeration and investigation of acute 0/1simplices 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), 213219.
Petr Lisonek, Combinatorial families enumerated by quasipolynomials, Journal of Combinatorial Theory, Series A, Volume 114, Issue 4, May 2007, Pages 619630.
Thomas Wieder, The number of certain kcombinations of an nset, 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(n1)  a(n3)  a(n4) + 2*a(n6)  a(n7) for n >= 8. [Harvey P. Dale, Dec 25 2011]


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 foursubgroup G of S_n such that the i^th largest (nontrivial) product of 2cycles in G consists of mu_i 2cycles, 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>floor(n*(2*n^2+21*n6)/72); seq(A034198(k), k=1..100); # Wesley Ivan Hurt, Oct 29 2013


MATHEMATICA

Table[Floor[n (2n^2+21*n6)/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*n6)/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



