A006381 - OEIS

%I M3313 #33 Mar 07 2025 11:49:16

%S 1,1,4,7,19,32,68,114,210,336,562,862,1349,1987,2950,4201,5991,8278,

%T 11422,15386,20660,27218,35718,46158,59401,75475,95494,119545,149035,

%U 184118,226562,276620,336470,406490,489344,585572,698397,828549,979896

%N Number of n X 3 binary matrices under row and column permutations and column complementations.

%C Also the number of ways in which to label the vertices of the cube (or faces of the octahedron) with nonnegative integers summing to n, where labelings that differ only by rotation or reflection are considered the same. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Andrew Howroyd, <a href="/A006381/b006381.txt">Table of n, a(n) for n = 0..1000</a>

%H M. A. Harrison, <a href="http://doi.org/10.1109/T-C.1973.223649">On the number of classes of binary matrices</a>, IEEE Trans. Computers, 22 (1973), 1048-1052.

%H <a href="/index/Rec#order_22">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,2,-5,7,-4,2,2,-8,10,-8,2,2,-4,7,-5,2,-1,-2,3,-1).

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%F G.f.: (1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48.

%F G.f.: (x^14 - 2*x^13 + 3*x^12 - 2*x^11 + 5*x^10 - 4*x^9 + 7*x^8 - 4*x^7 + 7*x^6 - 4*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 2*x + 1)/(x^6 - 1)/(x^2 + 1)^2/(x^2 + x + 1)/(x + 1)^3/(x - 1)^7.

%e Representatives of the seven classes of 3 X 3 binary matrices are:

%e [ 1 1 1 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 1 ] [ 0 1 1 ] [ 0 1 1 ] [ 0 1 1 ]

%e [ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 0 1 ] [ 1 0 0 ] [ 1 0 0 ]

%e [ 1 1 1 ] [ 1 1 1 ] [ 1 1 1 ] [ 1 1 0 ] [ 1 1 0 ] [ 1 1 1 ] [ 1 0 0 ].

%o (PARI) Vec((1/(1 - x^1)^8 + 13/(1 - x^2)^4 + 6/(1 - x^1)^4/(1 - x^2)^2 + 12/(1 - x^4)^2 + 8/(1 - x^1)^2/(1 - x^3)^2 + 8/(1 - x^2)^1/(1 - x^6)^1)/48 + O(x^41)) \\ _Andrew Howroyd_, May 30 2023

%Y Column k=3 of A363349.

%Y Cf. A000601, A005232, A006382, A006380, A002727, A006148.

%K nonn,nice,easy,changed

%O 0,3

%A _N. J. A. Sloane_

%E Entry revised by _Vladeta Jovovic_, Aug 05 2000

%E Definition corrected by _Max Alekseyev_, Feb 05 2010