A353933 a(n) is the permanent of the n X n symmetric matrix M(n) whose generic element M[i,j] is equal to the digital root of i*j.

1, 1, 8, 216, 7344, 168183, 7226091, 295259094, 11801772252, 1673511251940, 65568867621336, 2710049208604776, 202103867012027328, 12881755844526953376, 736186737257150962752, 70484099228399057425344, 5507570249593121504026368, 434305172863416192470350848, 122043063804581668929348667392

Offset: 0

Author

Stefano Spezia, May 11 2022

Keywords

• nonn
• base

Comments

The matrix M(n) is nonsingular only for n = 1, 5 and 6 with determinant equal respectively to 1, 6561 and 59049.

The rank of M(n) is 1 for 1 <= n <= 3, 3 for n = 4, 5 for n = 5, 6 for 6 <= n <= 8, and 7 for n >= 9. - Jianing Song, Sep 28 2022

Examples

			a(7) = 7226091:
     1, 2, 3, 4, 5, 6, 7
     2, 4, 6, 8, 1, 3, 5
     3, 6, 9, 3, 6, 9, 3
     4, 8, 3, 7, 2, 6, 1
     5, 1, 6, 2, 7, 3, 8
     6, 3, 9, 6, 3, 9, 6
     7, 5, 3, 1, 8, 6, 4
		

Crossrefs

Cf. A003991, A010888, A353109, A353128, A353974 (trace of the matrix M(n)).

Programs

Mathematica

M[i_, j_]:=If[i*j==0, 0, 1+Mod[i*j-1, 9]]; Join[{1},Table[Permanent[Table[M[i, j], {i,  n}, {j, n}]],{n,18}]]

PARI

a(n) = matpermanent(matrix(n, n, i, j, (i*j-1)%9+1)); \\ Michel Marcus, May 12 2022

Formula

Sum_{i=1..n} M[n-i+1,i] = A353128(n+1).