A353974 -id:A353974 - cp's OEIS frontend

A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 1, 3, 3, 1, 6, 0, 0, 7, 3, 6, 7, 6, 3, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 7, 3, 6, 7, 6, 3, 7, 9, 0, 0, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 0, 0, 2, 2, 9, 5, 8, 9, 8, 5, 9, 2, 2, 0

Offset: 0

Stefano Spezia, Apr 24 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 1, 2, 3, 4, 5, 6, 7, ...
    0, 2, 4, 6, 8, 1, 3, 5, ...
    0, 3, 6, 9, 3, 6, 9, 3, ...
    0, 4, 8, 3, 7, 2, 6, 1, ...
    0, 5, 1, 6, 2, 7, 3, 8, ...
    0, 6, 3, 9, 6, 3, 9, 6, ...
    0, 7, 5, 3, 1, 8, 6, 4, ...
    ...

Cf. A003991, A004247, A010888, A056992 (diagonal), A073636, A139413, A180592, A180593, A180594, A180595, A180596, A180597, A180598, A180599, A303296, A336225, A353128 (antidiagonal sums), A353933, A353974 (partial sum of the main diagonal).

A[i_,j_]:=If[i*j==0,0,1+Mod[i*j-1,9]];Flatten[Table[A[n-k,k],{n,0,12},{k,0,n}]]

T(n,k) = if (n && k, (n*k-1)%9+1, 0); \\ Michel Marcus, May 12 2022

A(n, k) = A010888(A004247(n, k)).

A(n, k) = A010888(A003991(n, k)) for n*k > 0.

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.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, May 11 2022

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

			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

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

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}]]

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

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

A357421 a(n) is the hafnian of the 2n X 2n symmetric matrix whose generic element M[i,j] is equal to the digital root of i*j.

Original entry on oeis.org

1, 2, 54, 1377, 55350, 4164534, 217595322, 11974135554, 999599777190, 150051627647010, 11873389098337236

Offset: 0

Stefano Spezia, Sep 27 2022

			a(3) = 1377:
    1, 2, 3, 4, 5, 6;
    2, 4, 6, 8, 1, 3;
    3, 6, 9, 3, 6, 9;
    4, 8, 3, 7, 2, 6;
    5, 1, 6, 2, 7, 3;
    6, 3, 9, 6, 3, 9.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A003991, A010888, A353109, A353933 (permanent of M(n)), A353974 (trace of M(n)).
Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356483, A356484, A357279.

M[i_, j_, n_] := If[i*j == 0, 0, 1 + Mod[i*j - 1, 9]]; a[n_] := Sum[Product[M[Part[PermutationList[s, 2 n], 2 i - 1], Part[PermutationList[s, 2 n], 2 i], 2 n], {i, n}], {s, SymmetricGroup[2 n] // GroupElements}]/(n!*2^n); Array[a, 6, 0]

a(6)-a(10) from Pontus von Brömssen, Oct 15 2023

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A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

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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.

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A357421 a(n) is the hafnian of the 2n X 2n symmetric matrix whose generic element M[i,j] is equal to the digital root of i*j.

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