A353933 -id:A353933 - 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.

A353128 Antidiagonal sums of A353109.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 20, 39, 48, 57, 40, 61, 58, 68, 92, 59, 96, 105, 114, 79, 118, 106, 116, 149, 98, 153, 162, 171, 118, 175, 154, 164, 206, 137, 210, 219, 228, 157, 232, 202, 212, 263, 176, 267, 276, 285, 196, 289, 250, 260, 320, 215, 324, 333, 342, 235, 346

Offset: 0

Stefano Spezia, Apr 24 2022

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

Cf. A004247, A010888, A353109, A353933.

LinearRecurrence[{0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1},{0,0,1,4,10,20,35,20,39,48,57,40,61,58,68,92,59,96,105},58]

G.f.: x^2*(1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4 + 20*x^5 + 39*x^6 + 48*x^7 + 55*x^9 + 32*x^10 + 41*x^11 + 18*x^12 - 2*x^13 - 19*x^15 + 105*x^17 + x^18 + 3*x^19 + 6*x^20 + 10*x^21 + 15*x^22 + 89*x^23 + 19*x^24 + 9*x^25 - 48*x^26)/((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2).

a(n) = 2*a(n-9) - a(n-18) for n > 18.

A353974 a(n) is the n-th partial sum of A056992.

Original entry on oeis.org

0, 1, 5, 14, 21, 28, 37, 41, 42, 51, 52, 56, 65, 72, 79, 88, 92, 93, 102, 103, 107, 116, 123, 130, 139, 143, 144, 153, 154, 158, 167, 174, 181, 190, 194, 195, 204, 205, 209, 218, 225, 232, 241, 245, 246, 255, 256, 260, 269, 276, 283, 292, 296, 297, 306, 307, 311

Offset: 0

Stefano Spezia, May 12 2022

Also the n-th partial sum of the main diagonal of A353109, or equivalently, the trace of the matrix M(n) whose permanent is A353933(n) for n > 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A003991, A010888, A056992, A071317, A353109, A353933.

CoefficientList[Series[x(1+4x+9x^2+7x^3+7x^4+9x^5+4x^6+x^7+9x^8)/((1-x)^2(1+x+x^2)(1+x^3+x^6)),{x,0,56}],x]

G.f.: x*(1 + 4*x + 9*x^2 + 7*x^3 + 7*x^4 + 9*x^5 + 4*x^6 + x^7 + 9*x^8)/((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)).
a(n) = a(n-1) + a(n-9) - a(n-10) for n > 9.
a(n) ~ 51*n/9.

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).

A362074 a(n) is the rank 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, 1, 3, 5, 6, 7, 7, 7, 7, 7, 9, 11, 11, 13, 14, 15, 16, 18, 18, 18, 19, 19, 21, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63

Offset: 1

Stefano Spezia, Apr 08 2023

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

Cf. A003991, A007953, A353933, A362072, A362073 (permanent).

M[i_, j_]:=Total[IntegerDigits[i*j]]; Table[MatrixRank[Table[M[i, j], {i,  n}, {j, n}]], {n, 69}]

a(n)=matrank(matrix(n,n,i,j,sumdigits(i*j))) \\ Andrew Howroyd, Apr 08 2023

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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A353128 Antidiagonal sums of A353109.

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A353974 a(n) is the n-th partial sum of A056992.

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

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A362074 a(n) is the rank of the n X n symmetric matrix M(n) whose generic element M[i,j] = digsum(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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