cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A211031 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant in the closed interval [-n,n].

Original entry on oeis.org

1, 16, 69, 176, 375, 650, 1107, 1626, 2413, 3326, 4527, 5782, 7689, 9436, 11753, 14354, 17491, 20458, 24623, 28334, 33425, 38438, 44031, 49450, 57323, 64028, 71849, 80078, 89857, 98468, 110545, 120388, 133117, 145382, 158699, 172256
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000, A211032.

Programs

  • Mathematica
    a = 0; 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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, -n, m}]
Table[c1[n, n], {n, 0, z1}]  (* A211031 *)
  • Python
    import numpy as np
def A211031_gen(limit):
    yield 1
    offset = limit + 1
    size = offset * offset + 1
    # a[offset+k] is the number of solutions to i*j = k with i,j in {0, 1, 2, ..., n}
    a = np.zeros(size, dtype=np.int64)
    a[offset] = 1
    for n in range(1, offset):
        a[offset: offset + n*n: n] += 2
        a[offset + n*n] += 1
        lag = 2*n + 1
        c = np.cumsum(a)
        c = c[lag:] - c[:-lag]
        a1 = a[n+1: -n]
        yield int(a1 @ c)
print(list(A211031_gen(35))) # David Radcliffe, Aug 15 2025

A211032 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant in the open interval (-n,n).

Original entry on oeis.org

10, 45, 134, 289, 560, 903, 1476, 2091, 3014, 4059, 5456, 6823, 8994, 10905, 13434, 16275, 19740, 22871, 27440, 31327, 36778, 42147, 48176, 53755, 62164, 69241, 77466, 86047, 96474, 105249, 118114, 128261, 141542, 154371, 168172
Offset: 1

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000, A211031.

Programs

  • Mathematica
    a = 0; 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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, -n, m}]
Table[c1[n, n - 1] - c1[n, -n], {n, 1, z1}] (* A211032 *)

A211033 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 0 (mod 3).

Original entry on oeis.org

1, 10, 33, 152, 297, 528, 1217, 1834, 2673, 4744, 6385, 8448, 13073, 16506, 20625, 29336, 35545, 42768, 57457, 67642, 79233, 102152, 117729, 135168, 168929, 191530, 216513, 264088, 295561, 330000, 394721, 437130, 483153, 568712, 624337, 684288, 794737, 866074
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

A211033(n) + 2*A211034(n)=n^4 for n>0. For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000, A211034.

Programs

  • Mathematica
    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, 0, z1}]    (* A211033 *)
Table[v[n], {n, 0, z1}]    (* A211034 *)
Table[w[n], {n, 0, z1}]    (* A211034 *)
  • Python
    from _future_ import division
def A211033(n):
    x,y,z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
    return x**4 + 4*x**3*y + 4*x**3*z + 4*x**2*y**2 + 8*x**2*y*z + 4*x**2*z**2 + y**4 + 6*y**2*z**2 + z**4 # Chai Wah Wu, Nov 28 2016

Formula

From Chai Wah Wu, Nov 28 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 > 12.
G.f.: (-x^11 - 7*x^10 - 25*x^9 - 53*x^8 - 91*x^7 - 219*x^6 - 139*x^5 - 109*x^4 - 115*x^3 - 23*x^2 - 9*x - 1)/((x - 1)^5*(x^2 + x + 1)^4).
If r = floor(n/3)+1, s = floor((n-1)/3)+1 and t = floor((n-2)/3)+1, then:
a(n) = r^4 + 4*r^3*s + 4*r^3*t + 4*r^2*s^2 + 8*r^2*s*t + 4*r^2*t^2 + s^4 + 6*s^2*t^2 + t^4.
If n == 0 mod 3, then a(n) = (11*n^4 + 60*n^3 + 138*n^2 + 108*n)/27 + 1.
If n == 1 mod 3, then a(n) = (11*n^4 + 52*n^3 + 96*n^2 + 76*n + 35)/27.
If n == 2 mod 3, then a(n) = 11*(n + 1)^4/27. (End)

A211034 Number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 1 (mod 3).

Original entry on oeis.org

0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744, 10955, 15000, 18100, 23988, 31104, 36432, 46179, 57624, 66052, 81056, 98304, 110848, 132723, 157464, 175284, 205860, 240000, 264400, 305723, 351384, 383812, 438144, 497664, 539712, 609531
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2012

Keywords

Comments

Also, the number of 2 X 2 matrices having all elements in {0,1,...,n} and determinant = 2 (mod 3). A211033(n) + 2*A211034(n)=n^4 for n>0. For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000, A211033.

Programs

  • Mathematica
    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, 0, z1}]   (* A211033 *)
Table[v[n], {n, 0, z1}]   (* A211034 *)
Table[w[n], {n, 0, z1}]   (* A211034 *)
LinearRecurrence[{1, 0, 4, -4, 0, -6, 6, 0, 4, -4, 0, -1, 1}, {0, 3, 24, 52, 164, 384, 592, 1131, 1944, 2628, 4128, 6144, 7744}, 60] (* Vincenzo Librandi, Nov 29 2016 *)
  • Python
    from _future_ import division
def A211034(n):
    x,y,z = n//3 + 1, (n-1)//3 + 1, (n-2)//3 + 1
    return x**2*y**2 + 2*x**2*y*z + x**2*z**2 + 2*x*y**3 + 6*x*y**2*z + 6*x*y*z**2 + 2*x*z**3 + 2*y**3*z + 2*y*z**3 # Chai Wah Wu, Nov 28 2016

Formula

From Chai Wah Wu, Nov 28 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 > 12.
G.f.: -x*(4*x^9 + 20*x^8 + 59*x^7 + 109*x^6 + 96*x^5 + 136*x^4 + 100*x^3 + 28*x^2 + 21*x + 3)/((x - 1)^5*(x^2 + x + 1)^4).
If r = floor(n/3)+1, 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) = 4*n^2*(2*n^2 + 6*n + 3)/27.
If n == 1 mod 3, then a(n) = (8*n^4 + 28*n^3 + 33*n^2 + 16*n - 4)/27.
If n == 2 mod 3, then a(n) = 8*(n + 1)^4/27. (End)

A211053 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant n.

Original entry on oeis.org

0, 2, 7, 16, 23, 50, 45, 93, 99, 150, 117, 283, 167, 308, 336, 443, 289, 654, 369, 803, 658, 762, 543, 1392, 779, 1092, 1086, 1563, 879, 2160, 1011, 2038, 1652, 1888, 1758, 3323, 1445, 2386, 2302, 3730, 1795, 4220, 1989, 3889, 3737
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Examples

			a(2) counts these 2 matrices:
2 1.....2 2
2 2.....1 2

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    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]
Table[c[n, n], {n, 1, z1}]  (* A211053 *)

A211065 Number of 2 X 2 matrices having all terms in {1,...,n} and odd determinant.

Original entry on oeis.org

0, 6, 40, 96, 288, 486, 1056, 1536, 2800, 3750, 6120, 7776, 11760, 14406, 20608, 24576, 33696, 39366, 52200, 60000, 77440, 87846, 110880, 124416, 154128, 171366, 208936, 230496, 277200, 303750, 360960, 393216, 462400, 501126, 583848
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

A211064(n)+A211065(n)=4^n.
For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    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]
u[n_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 1, z1}] (* A211064 *)
Table[v[n], {n, 1, z1}] (* A211065 *)

Formula

From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 1 -(-1)^n)*(2*n - 1 + (-1)^n)/128.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: -2*x^2*(3*x^5 + 5*x^4 + 28*x^3 + 16*x^2 + 17*x + 3)/((x - 1)^5*(x + 1)^4).
(End)

A211066 Number of 2 X 2 matrices having all terms in {1,...,n} and nonnegative even determinant.

Original entry on oeis.org

1, 8, 28, 96, 193, 448, 728, 1360, 1985, 3264, 4420, 6696, 8641, 12296, 15360, 20896, 25361, 33344, 39636, 50680, 59289, 74056, 85376, 104728, 119377, 144032, 162588, 193568, 216585, 254880, 283096, 329656, 363881, 419856, 460804
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    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]
u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
Table[u[n], {n, 1, z1}]  (* A211066 *)
Table[v[n], {n, 1, z1}]  (* A211067 *)
Table[w[n], {n, 1, z1}]  (* A211068 *)

Formula

a(n) = (A211064(n) + A134506(n))/2. - Chai Wah Wu, Nov 28 2016

A211067 Number of 2 X 2 matrices having all terms in {1,...,n} and positive even determinant.

Original entry on oeis.org

0, 2, 13, 64, 144, 362, 617, 1200, 1776, 2986, 4101, 6264, 8160, 11714, 14657, 20064, 24464, 32266, 38485, 49320, 57752, 72354, 83585, 102632, 117120, 141578, 159917, 190592, 213496, 251370, 279465, 325704, 359640, 415354, 455973
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    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]
u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
Table[u[n], {n, 1, z1}]  (* A211066 *)
Table[v[n], {n, 1, z1}]  (* A211067 *)
Table[w[n], {n, 1, z1}]  (* A211068 *)

Formula

a(n) = (A211064(n) - A134506(n))/2. - Chai Wah Wu, Nov 28 2016

A211068 Number of 2 X 2 matrices having all terms in {1,...,n} and positive odd determinant.

Original entry on oeis.org

0, 3, 20, 48, 144, 243, 528, 768, 1400, 1875, 3060, 3888, 5880, 7203, 10304, 12288, 16848, 19683, 26100, 30000, 38720, 43923, 55440, 62208, 77064, 85683, 104468, 115248, 138600, 151875, 180480, 196608, 231200, 250563, 291924
Offset: 1

Views

Author

Clark Kimberling, Mar 31 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000.

Programs

  • Magma
    [(2*n+1-(-1)^n)^2*(6*n+1-(-1)^n)*(2*n-1+(-1)^n)/256: n in [1..40]]; // Vincenzo Librandi, Nov 28 2016
  • Mathematica
    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]
u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
Table[u[n], {n, 1, z1}]  (* A211066 *)
Table[v[n], {n, 1, z1}]  (* A211067 *)
Table[w[n], {n, 1, z1}]  (* A211068 *)
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 3, 20, 48, 144, 243, 528, 768, 1400}, 50] (* Vincenzo Librandi, Nov 28 2016 *)

Formula

From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = A211065(n)/2.
a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 1 -(-1)^n)*(2*n - 1 + (-1)^n)/256.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: -x^2*(3*x^5 + 5*x^4 + 28*x^3 + 16*x^2 + 17*x + 3)/((x - 1)^5*(x + 1)^4). (End)

A211155 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and odd determinant.

Original entry on oeis.org

0, 40, 168, 1056, 2080, 6120, 9576, 20608, 28800, 52200, 68200, 110880, 138528, 208936, 252840, 360960, 426496, 583848, 677160, 896800, 1024800, 1321320, 1491688, 1881216, 2102400, 2602600, 2883816, 3513888, 3865120, 4645800, 5077800, 6031360, 6555648, 7705896, 8334760, 9707040
Offset: 0

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

A211154(n) + A211155(n) = (2n+1)^4.
For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000.

Programs

  • Maple
    seq( 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n), n=1..20); # Mark van Hoeij, May 13 2013
  • Mathematica
    a = -n; b = n; z1 = 20;
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_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 1, z1}] (* A211154 *)
Table[v[n], {n, 1, z1}] (* A211155 *)
  • PARI
    a(n)=2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n); \\ Joerg Arndt, May 14 2013

Formula

From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(-40*x^6 - 128*x^5 - 728*x^4 - 512*x^3 - 728*x^2 - 128*x - 40)/((x - 1)^5*(x + 1)^4). (End)

Extensions

More terms from Joerg Arndt, May 14 2013
a(0)=0 prepended by Andrew Howroyd, May 05 2020
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