|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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]
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.
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):
|
|
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
|
Column k=2 of both A034356 and A076831 (which are the same except for column k=0).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|