A059976 -id:A059976 - cp's OEIS frontend

A064276 Number of 2 X 2 singular integer matrices with elements from {0,...,n} up to row and column permutation.

1, 5, 13, 25, 42, 62, 90, 118, 155, 195, 243, 287, 352, 404, 472, 548, 629, 697, 797, 873, 986, 1094, 1202, 1294, 1443, 1559, 1687, 1823, 1984, 2100, 2296, 2420, 2597, 2769, 2937, 3125, 3366, 3514, 3702, 3906, 4167, 4331, 4611, 4783, 5040, 5320, 5548

Offset: 0

Vladeta Jovovic, Sep 24 2001

nonn

			There are 5 binary singular matrices up to row and column permutation:
[0 0] [1 0] [1 1] [1 0] [1 1]
[0 0] [0 0] [0 0] [1 0] [1 1].

Cf. A059306, A062801, A059976.

a(n) = (A059306(n)+(n+1)*(2*n+3))/4.

More terms from David Wasserman, Jul 16 2002

A064368 Number of 2 X 2 symmetric singular matrices with entries from {0,...,n}.

Original entry on oeis.org

1, 4, 7, 10, 15, 18, 21, 24, 29, 36, 39, 42, 47, 50, 53, 56, 65, 68, 75, 78, 83, 86, 89, 92, 97, 108, 111, 118, 123, 126, 129, 132, 141, 144, 147, 150, 163, 166, 169, 172, 177, 180, 183, 186, 191, 198, 201, 204, 213, 228, 239, 242, 247, 250, 257, 260, 265, 268

Offset: 0

Vladeta Jovovic, Sep 27 2001

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000010, A000188, A064276, A059306, A062801, A059976, A039623.

Cf. A001620, A306016.

f[p_, e_] := p^Floor[e/2]; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 100}, 1 + Range[0, max] + 2 * Accumulate[Array[a, max + 1, 0]]] (* Amiram Eldar, Nov 07 2024 *)

a(n) = n + 1 + 2*sum(k=1, n, sumdiv(k, d, issquare(d)*eulerphi(sqrtint(d)))) \\ Michel Marcus, Jun 17 2013

a(n) = n + 1 + 2*Sum_{k=1..n} Sum_{d^2|k} phi(d), where phi = Euler totient function A000010.

a(n) ~ (n/zeta(2)) * (log(n) + 3*gamma - 1 + zeta(2) - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 07 2024

A277636 Number of 3 X 3 matrices having all elements in {0,...,n} with determinant = permanent.

Original entry on oeis.org

1, 343, 6859, 50653, 226981, 753571, 2048383, 4826809, 10218313, 19902511, 36264691, 62570773, 103161709, 163667323, 251239591, 374805361, 545338513, 776151559, 1083206683, 1485446221, 2005142581, 2668267603, 3504881359, 4549540393, 5841725401, 7426288351

Offset: 0

Indranil Ghosh, Jan 02 2017

a(n) is a perfect cube.

Indranil Ghosh, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A059976 (Number of 3 X 3 singular matrices with all elements in {0,...,n})
Cf. A015237 (Number of 2 X 2 matrices with all elements in {0,...,n} with determinant = permanent )
Cf. A003215.

Vec((1 + 336*x + 4479*x^2 + 9808*x^3 + 4479*x^4 + 336*x^5 + x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jan 02 2017

def a(n):
    return 27*n**6-81*n**5+108*n**4-81*n**3+36*n**2-9*n+1

a(n) = A003215(n-1)^3.
a(n) = (3*n^2 - 3*n + 1)^3.
G.f.: (1 + 336*x + 4479*x^2 + 9808*x^3 + 4479*x^4 + 336*x^5 + x^6) / (1 - x)^7. - Colin Barker, Jan 02 2017

A064363 Number of 2 X 2 regular integer matrices with elements from {0,...,n} up to row and column permutation.

Original entry on oeis.org

0, 2, 14, 51, 133, 289, 547, 954, 1546, 2380, 3508, 5005, 6915, 9347, 12353, 16028, 20468, 25790, 32054, 39427, 47965, 57833, 69155, 82082, 96682, 113192, 131720, 152429, 175467, 201075, 229305, 260492, 294700, 332182, 373138, 417751, 466201

Offset: 0

Vladeta Jovovic, Sep 25 2001

			There are 2 binary regular matrices up to row and column permutation:
[1 0] [1 1]
[0 1] [1 0].

Cf. A064276, A059306, A062801, A059976, A039623.

A059306[0] = 1; A059306[n_] := Table[{w, x, y, z} /. {ToRules[ Reduce[0 <= x <= n && 0 <= y <= n && 0 <= z <= n && w*z - x*y == 0, {x, y, z}, Integers]]}, {w, 0, n}] // Flatten[#, 1] & // Length; a[n_] := ((n + 1)*(n^3 + 3*n^2 + 4*n + 1) - A059306[n])/4; Table[Print[an = a[n]]; an, {n, 0, 36}] (* Jean-François Alcover, Nov 26 2013 *)

a(n) = ((n+1)*(n^3+3*n^2+4*n+1)-A059306(n))/4.

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A064276 Number of 2 X 2 singular integer matrices with elements from {0,...,n} up to row and column permutation.

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A064368 Number of 2 X 2 symmetric singular matrices with entries from {0,...,n}.

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A277636 Number of 3 X 3 matrices having all elements in {0,...,n} with determinant = permanent.

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A064363 Number of 2 X 2 regular integer matrices with elements from {0,...,n} up to row and column permutation.

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