A210000 -id:A210000 - cp's OEIS frontend

A211062 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >= 2n.

0, 0, 5, 29, 93, 235, 493, 936, 1622, 2642, 4057, 6055, 8608, 12001, 16297, 21619, 28127, 36170, 45624, 57041, 70350, 85892, 103748, 124748, 148174, 175083, 205683, 239800, 277908, 321224, 368301, 421093, 479209, 543193, 613374

Offset: 1

Clark Kimberling, Mar 31 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A210000.

a = 1; b = n; z1 = 40;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
d[n_, m_] := d[n, m] = Sum[c[n, k], {k, 0, m}]
Table[d[n, n^2] - d[n, 2 n - 1], {n, 1, z1}]

A211063 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >=3n.

Original entry on oeis.org

0, 0, 0, 0, 8, 38, 122, 284, 589, 1080, 1848, 2933, 4515, 6577, 9366, 12888, 17444, 22928, 29864, 37986, 48057, 59656, 73413, 89307, 108002, 129068, 153386, 180636, 212111, 246683, 286126, 329135, 378047, 431242, 490425, 555118

Offset: 0

Clark Kimberling, Mar 31 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 0..2000

Cf. A210000.

a = 1; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
s[n_, m_] := s[n, m] = Sum[c[n, k], {k, 1, m}]
Table[s[n,n^2]-s[n,3n-1],{n,1,z1}] (* A211063 *)

Offset corrected by Chai Wah Wu, Jan 10 2017

A211071 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 1 (mod 3).

Original entry on oeis.org

0, 4, 24, 83, 208, 384, 756, 1332, 1944, 3099, 4672, 6144, 8768, 12100, 15000, 19995, 26064, 31104, 39588, 49588, 57624, 70931, 86272, 98304, 117984, 140292, 157464, 185283, 216400, 240000, 277940, 319924, 351384, 401643, 456768

Offset: 1

Clark Kimberling, Apr 01 2012

nonn

Also, the number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 2 (mod 3).
A210698(n) + 2*A211071(n) = n^4.
For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1).

Cf. A210000, A210698, A211034.

a = 1; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
u[n_] := u[n] = Sum[c[n, 3 k], {k, -2*n^2, 2*n^2}]
v[n_] := v[n] = Sum[c[n, 3 k + 1], {k, -2*n^2, 2*n^2}]
w[n_] := w[n] = Sum[c[n, 3 k + 2], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 1, z1}] (* A210698 *)
Table[v[n], {n, 1, z1}] (* A211071 *)
Table[w[n], {n, 1, z1}] (* A211071 *)
LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {0, 4, 24, 83, 208, 384, 756, 1332, 1944, 3099, 4672, 6144, 8768}, 40] (* Vincenzo Librandi, Dec 01 2016 *)

from _future_ import division
def A211071(n):
    if n % 3 == 0:
        return 8*n**4//27
    elif n % 3 == 1:
        return (8*n**4 + 4*n**3 - 3*n**2 - 2*n - 7)//27
    else:
        return (8*n**4 + 8*n**3 - 12*n**2 - 16*n - 4)//27 # Chai Wah Wu, Nov 30 2016

From Chai Wah Wu, Nov 30 2016: (Start)
a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4) - 6*a(n-6) + 6*a(n-7) + 4*a(n-9) - 4*a(n-10) - a(n-12) + a(n-13) for n > 13.
G.f.: -x^2*(3*x^9 + 21*x^8 + 28*x^7 + 100*x^6 + 136*x^5 + 96*x^4 + 109*x^3 + 59*x^2 + 20*x + 4)/((x - 1)^5*(x^2 + x + 1)^4).
If r = floor(n/3), s = floor((n-1)/3)+1 and t = floor((n-2)/3)+1, then:
a(n) = r^2*s^2 + 2*r^2*s*t + r^2*t^2 + 2*r*s^3 + 6*r*s^2*t + 6*r*s*t^2 + 2*r*t^3 + 2*s^3*t + 2*s*t^3.
If n == 0 mod 3, then a(n) = 8*n^4/27.
If n == 1 mod 3, then a(n) = (8*n^4 + 4*n^3 - 3*n^2 - 2*n - 7)/27.
If n == 2 mod 3, then a(n) = (8*n^4 + 8*n^3 - 12*n^2 - 16*n - 4)/27. (End)

A211140 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant n.

Original entry on oeis.org

1, 20, 92, 176, 364, 408, 880, 704, 1420, 1412, 2088, 1552, 3760, 2104, 3808, 4096, 5388, 3400, 7660, 4208, 9160, 7392, 8464, 5952, 15344, 8428, 11656, 11584, 16608, 9208, 22752, 10464, 21132, 16928, 19192, 17952

Offset: 0

Clark Kimberling, Apr 02 2012

nonn

For a guide to related sequences, see A210000.

Also the number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and permanent n [because #(a,b,c,d) with a,b,c,d in {-n..n} and a*d-b*c=n equals #(a,b,c,d) with a,b,c,d in {-n..n} and a*d+b*c=n. (Replace d with -d)]. - Alois P. Heinz, Jun 26 2012

Chai Wah Wu, Table of n, a(n) for n = 0..3000

Cf. A210000.

a = -n; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n], {n, 0, z1}]  (* A211140 *)

A211141 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant n-1.

Original entry on oeis.org

0, 33, 52, 156, 240, 492, 472, 1072, 832, 1612, 1540, 2408, 1680, 4144, 2296, 4064, 4352, 5900, 3592, 8236, 4464, 9544, 7712, 9168, 6208, 15984, 8812, 12232, 11968, 17504, 9464, 23712, 10976, 21772, 17440, 19960

Offset: 0

Clark Kimberling, Apr 02 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 0..3000

Cf. A210000.

a = -n; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n - 1], {n, 0, z1}]  (* A211141 *)

A211142 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant n+1.

Original entry on oeis.org

0, 4, 24, 116, 128, 408, 296, 788, 748, 1232, 888, 2488, 1312, 2568, 2664, 3860, 2352, 5652, 3032, 6864, 5320, 6456, 4520, 12152, 6356, 9264, 9000, 13288, 7392, 18440, 8520, 17556, 13576, 16032, 14216, 28612

Offset: 0

Clark Kimberling, Apr 03 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 0..3000

Cf. A210000.

a = -n; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n + 1], {n, 0, z1}]   (* A211142 *)
%/4                              (* integers *)

A211143 Number of 2 X 2 matrices having all terms in {-n, ..., 0, ..., n} and determinant = n^2.

Original entry on oeis.org

1, 20, 52, 84, 132, 156, 260, 228, 356, 372, 492, 404, 804, 508, 820, 844, 964, 716, 1396, 852, 1660, 1380, 1540, 1092, 2452, 1476, 1932, 1876, 2564, 1532, 3884, 1700, 3012, 2676, 3004, 2876, 4916, 2172, 3684, 3484, 5260

Offset: 0

Clark Kimberling, Apr 03 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 0..3000

Cf. A210000.

a = -n; b = n; z1 = 40;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, n^2], {n, 0, z1}]         (* A211143 *)
(1/4) Table[c[n, n^2], {n, 1, z1}]   (* integers *)

A211146 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and 0 <= determinant <= n.

Original entry on oeis.org

1, 53, 273, 737, 1613, 2821, 4853, 7125, 10593, 14597, 19885, 25309, 33677, 41189, 51269, 62565, 76145, 88793, 106821, 122581, 144541, 166045, 189997, 212877, 246653, 275081, 308369, 343281, 384977, 421097, 472649, 513865, 567765

Offset: 0

Clark Kimberling, Apr 03 2012

nonn

For a guide to related sequences, see A210000.

Cf. A210000.

a = -n; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n], {n, 0, z1}]              (* A211146 *)
(1/4) (-1 + Table[c1[n, n], {n, 0, z1}]) (* integers *)

Offset changed to 0 by Georg Fischer, Feb 02 2022

A211150 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant > n.

Original entry on oeis.org

4, 104, 608, 1940, 4916, 10084, 19052, 32352, 52084, 79308, 116900, 164564, 227860, 306324, 403868, 522256, 667488, 837236, 1041708, 1277060, 1553116, 1871084, 2238452, 2648836, 3121648, 3652200, 4248656, 4911312, 5656784

Offset: 1

Clark Kimberling, Apr 04 2012

nonn

For a guide to related sequences, see A210000.

Cf. A210000.

a = -n; b = n; z1 = 30;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, -2*n^2, m}]
Table[c1[n, 2*n^2] - c1[n, n], {n, 1, z1}]  (* A211150 *)
%/4  (* integers *)

A211151 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant > n.

Original entry on oeis.org

24, 196, 784, 2304, 5324, 10964, 19756, 33772, 53496, 81396, 118452, 168324, 229964, 310132, 407964, 527644, 670888, 844896, 1045916, 1286220, 1560508, 1879548, 2244404, 2664180, 3130076, 3663856, 4260240, 4927920, 5665992

Offset: 1

Clark Kimberling, Apr 04 2012

nonn

For a guide to related sequences, see A210000.

Cf. A210000.

a = -n; b = n; z1 = 30;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, -2*n^2, m}]
Table[c1[n,2*n^2]-c1[n,n-1],{n,1,z1}] (*A211151*)
%/4  (* integers *)

cp's OEIS Frontend

A211062 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >= 2n.

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A211063 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >=3n.

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A211071 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 1 (mod 3).

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A211140 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant n.

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A211141 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant n-1.

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A211142 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant n+1.

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A211143 Number of 2 X 2 matrices having all terms in {-n, ..., 0, ..., n} and determinant = n^2.

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A211146 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and 0 <= determinant <= n.

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Extensions

A211150 Number of 2 X 2 matrices having all terms in {-n,...,0,...,n} and determinant > n.

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