A336225 -id:A336225 - cp's OEIS frontend

A362072 Antidiagonal sums of A336225.

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

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

Original entry on oeis.org

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.

A362073 a(n) is the permanent of the n X n symmetric matrix M(n) whose generic element M[i,j] = digsum(i*j).

Original entry on oeis.org

1, 1, 8, 216, 7344, 168183, 7226091, 506792295, 43261224876, 5520748306176, 170835815638728, 19632554202684096, 2228687316428293152, 347514692118635694888, 62201193604462666921968, 8113764691750577654439864, 1557556394182730485102253088, 348394812690307787609428395792

Offset: 0

Stefano Spezia, Apr 08 2023

The matrix M(n) is nonsingular only for n = 1, 5, 6 and 7 with determinant equal respectively to 1, 6561, 59049 and -531441.

			a(6) = 7226091:
    [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]

Cf. A003991, A007953, A336225, A353933, A362072, A362074 (rank).

M[i_, j_]:=Total[IntegerDigits[i*j]]; Join[{1}, Table[Permanent[Table[M[i, j], {i,  n}, {j, n}]], {n, 18}]]

a(n) = matpermanent(matrix(n, n, i, j, sumdigits(i*j))); \\ Michel Marcus, Apr 08 2023

Sum_{i=1..n} M[n-i+1,i] = A362072(n).

cp's OEIS Frontend

A362072 Antidiagonal sums of A336225.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A362073 a(n) is the permanent of the n X n symmetric matrix M(n) whose generic element M[i,j] = digsum(i*j).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula