|
%I #6 Jul 12 2012 00:39:59
%S 1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T 2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%U 3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4
%N Symmetric matrix: f(i,j)=floor[(2i+2j+1)/4]-floor[(i+j)/4], by (constant) antidiagonals.
%C For n>=1, the number of occurrences of n is 16n-6. In the following guide to related matrices and permanents, Duvwxyz represents the matrix remaining after row 1 of the matrix Auvwxyz is deleted:
%C Matrix...............Permanent of n-th submatrix
%C A204551..............A204552
%C A204553=D204551......A204554
%C A204560=D204553......A204561
%C A204562=D204560......A204563
%e Northwest corner:
%e 1 1 1 1 2 2 2 2 3
%e 1 1 1 2 2 2 2 3 3
%e 1 1 2 2 2 2 3 3 3
%e 1 2 2 2 2 3 3 3 3
%e 2 2 2 2 3 3 3 3 4
%e 2 2 2 3 3 3 3 4 4
%e 2 3 3 3 3 4 4 4 4
%t f[i_, j_] :=
%t Floor[(2 i + 2 j + 1)/4] - Floor[(i + j)/4];
%t m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t TableForm[m[16]] (* 8x8 principal submatrix *)
%t Flatten[Table[f[i, n + 1 - i],
%t {n, 1, 14}, {i, 1, n}]] (* A204551 *)
%t Permanent[m_] :=
%t With[{a = Array[x, Length[m]]},
%t Coefficient[Times @@ (m.a), Times @@ a]];
%t Table[Permanent[m[n]], {n, 1, 15}] (* A204552 *)
%Y Cf. A204551, A204296.
%K nonn,tabl
%O 1,11
%A _Clark Kimberling_, Jan 16 2012
|
