A353128 -id:A353128 - 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).

A362072 Antidiagonal sums of A336225.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 20, 39, 48, 57, 58, 61, 58, 95, 110, 140, 114, 159, 186, 205, 172, 160, 134, 203, 206, 252, 216, 288, 262, 265, 280, 281, 260, 371, 354, 381, 282, 382, 430, 454, 410, 425, 392, 528, 510, 528, 466, 568, 628, 593, 626, 629, 594, 666, 684, 775

Offset: 0

Stefano Spezia, Apr 08 2023

Stefano Spezia, Table of n, a(n) for n = 0..10000

Cf. A003991, A007953, A336225, A353128.

T[i_, j_]:=Total[IntegerDigits[i*j]]; Table[Sum[T[n-j, j],{j,0,n}],{n,0,56}]

a(n) = sum(k=1, n, sumdigits(k*(n-k))) \\ Andrew Howroyd, Apr 08 2023

a(n) = Sum_{k=1..n} A007953(k*(n-k)). - Andrew Howroyd, Apr 08 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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A362072 Antidiagonal sums of A336225.

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