A353109 -id:A353109 - cp's OEIS frontend

A353128 Antidiagonal sums of A353109.

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.

A180597 Digital root of 7n.

Original entry on oeis.org

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

Offset: 0

Odimar Fabeny, Sep 10 2010

Period of 9. - Robert G. Wilson v, Sep 20 2010

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

Cf. A010888, A008589.
Cf. A180592, A180594, A180595, A180596, A180592, A180593, A180594, A180595, A180596, A180598, A180599.
Column k = 7 in A353109.

f[n_] := Mod[7 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)

a(n)=7*n%9 \\ Charles R Greathouse IV, Aug 22 2014

G.f.: x*(7 + 5*x + 3*x^2 + x^3 + 8*x^4 + 6*x^5 + 4*x^6 + 2*x^7 + 9*x^8)/(1 - x^9). - Stefano Spezia, Apr 21 2022

a(n) = A010888(A008589(n)). - Michel Marcus, Apr 21 2022

More terms from Robert G. Wilson v, Sep 20 2010

A180596 Digital root of 6n.

Original entry on oeis.org

0, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3

Offset: 0

Odimar Fabeny, Sep 10 2010

Period of 3. - Robert G. Wilson v, Sep 20 2010

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

Cf. A010888, A008588.
Cf. A180592, A180593, A180594, A180595, A180597, A180598, A180599.
Column k = 6 in A353109.

f[n_] := Mod[6 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
PadRight[{0},120,{9,6,3}] (* Harvey P. Dale, Dec 18 2012 *)

G.f.: 3*x*(2 + x + 3*x^2)/(1 - x^3). - Stefano Spezia, Apr 21 2022

a(n) = A010888(A008588(n)). - Michel Marcus, Apr 24 2022

More terms from Robert G. Wilson v, Sep 20 2010

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).

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.

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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A353128 Antidiagonal sums of A353109.

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A180597 Digital root of 7n.

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A180596 Digital root of 6n.

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

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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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