A210000 -id:A210000 - cp's OEIS frontend

A210366 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant >=n.

1, 3, 18, 61, 168, 368, 749, 1310, 2235, 3493, 5291, 7640, 10869, 14711, 19860, 26051, 33623, 42618, 53725, 66280, 81577, 98739, 118847, 141800, 168435, 197406, 231100, 268595, 310617, 356763, 409136, 465231, 528830, 597397, 673127

Offset: 0

Clark Kimberling, Mar 20 2012

nonn

See A210000 for a guide to related sequences.

Cf. A210000.

a = 0; 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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n^2] - c1[n, n - 1], {n, 0, z1}](* A210366 *)

A210367 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant >= 2n.

Original entry on oeis.org

1, 0, 5, 26, 83, 203, 456, 853, 1497, 2477, 3860, 5690, 8305, 11470, 15684, 20947, 27328, 35057, 44569, 55569, 68849, 84153, 101912, 122125, 146014, 172246, 202378, 236593, 274399, 316494, 364473, 416055, 473938, 537432, 607225, 683593, 767218, 857111, 955830

Offset: 0

Clark Kimberling, Mar 20 2012

nonn

See A210000 for a guide to related sequences.

			a(2)=5 counts these matrices:
  2 0...2 1...2 0...2 0...2 2
  0 2...0 2...1 2...2 2...0 2

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A210000.

a = 0; 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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n^2] - c1[n, 2 n - 1], {n, 0, z1}]

A210368 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant >= 3n.

Original entry on oeis.org

1, 0, 0, 7, 35, 104, 252, 509, 997, 1688, 2751, 4175, 6240, 8872, 12324, 16639, 22196, 28668, 36931, 46332, 58184, 71524, 87183, 105334, 126524, 150233, 177448, 207751, 242948, 281139, 324739

Offset: 0

Clark Kimberling, Mar 20 2012

nonn

See A210000 for a guide to related sequences.

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

Cf. A210000.

a = 0; 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, 0, m}]
Table[c1[n, n^2] - c1[n, 3 n - 1], {n, 0, z1}]

A210375 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 3.

Original entry on oeis.org

0, 1, 16, 44, 80, 125, 180, 246, 324, 415, 520, 640, 776, 929, 1100, 1290, 1500, 1731, 1984, 2260, 2560, 2885, 3236, 3614, 4020, 4455, 4920, 5416, 5944, 6505, 7100, 7730, 8396, 9099, 9840, 10620, 11440, 12301, 13204, 14150, 15140, 16175, 17256, 18384, 19560

Offset: 0

Clark Kimberling, Mar 20 2012

A210375 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3*n - 3.

See A210000 for a guide to related sequences.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A210000.

a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w + x + y + z, {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 + 3], {n, 0, z1}]   (* A210375 *)
Table[c[n, 3 n - 3], {n, 0, z1}] (* A210375 *)

From Colin Barker, Dec 07 2017: (Start)
G.f.: x*(1 + 12*x - 14*x^2 - 4*x^3 + 6*x^4) / (1 - x)^4.
a(n) = (-120 + 74*n + 15*n^2 + n^3) / 6 for n > 1.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 5. (End)

A210377 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 5.

Original entry on oeis.org

0, 0, 4, 31, 80, 146, 224, 315, 420, 540, 676, 829, 1000, 1190, 1400, 1631, 1884, 2160, 2460, 2785, 3136, 3514, 3920, 4355, 4820, 5316, 5844, 6405, 7000, 7630, 8296, 8999, 9740, 10520, 11340, 12201, 13104, 14050, 15040, 16075, 17156, 18284

Offset: 0

Clark Kimberling, Mar 20 2012

nonn

A210376 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3n - 5.

See A210000 for a guide to related sequences.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A210000.

a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w + x + y + z, {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 + 5], {n, 0, z1}]    (* A210377 *)
Table[c[n, 3 n - 5], {n, 0, z1}]  (* A210377 *)

Conjectures from Colin Barker, Dec 07 2017: (Start)
G.f.: x^2*(4 + 15*x - 20*x^2 - 4*x^3 + 6*x^5) / (1 - x)^4.
a(n) = (-504 + 146*n + 21*n^2 + n^3) / 6 for n>3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>5.
(End)

A210698 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 0 (mod 3).

Original entry on oeis.org

1, 8, 33, 90, 209, 528, 889, 1432, 2673, 3802, 5297, 8448, 11025, 14216, 20625, 25546, 31393, 42768, 51145, 60824, 79233, 92394, 107297, 135168, 154657, 176392, 216513, 244090, 274481, 330000, 367641, 408728, 483153, 533050, 587089

Offset: 1

Clark Kimberling, Apr 01 2012

nonn

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, A211071, A211033.

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}, {1, 8, 33, 90, 209, 528, 889, 1432, 2673, 3802, 5297, 8448, 11025}, 40] (* Vincenzo Librandi, Dec 01 2016 *)

from _future_ import division
def A210698(n):
    if n % 3 == 0:
        return 11*n**4//27
    elif n % 3 == 1:
        return (11*n**4 - 8*n**3 + 6*n**2 + 4*n + 14)//27
    else:
        return (11*n**4 - 16*n**3 + 24*n**2 + 32*n + 8)//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*(-x^11 - 9*x^10 - 23*x^9 - 115*x^8 - 109*x^7 - 139*x^6 - 219*x^5 - 91*x^4 - 53*x^3 - 25*x^2 - 7*x - 1)/((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^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^4/27.
If n == 1 mod 3, then a(n) = (11*n^4 - 8*n^3 + 6*n^2 + 4*n + 14)/27.
If n == 2 mod 3, then a(n) = (11*n^4 - 16*n^3 + 24*n^2 + 32*n + 8)/27. (End)

A211054 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant n-1.

Original entry on oeis.org

0, 1, 2, 6, 11, 26, 27, 66, 57, 109, 109, 182, 129, 319, 187, 332, 360, 497, 309, 714, 397, 839, 690, 842, 571, 1456, 821, 1156, 1126, 1663, 907, 2264, 1071, 2106, 1708, 1976, 1798, 3449, 1513, 2474, 2358, 3874, 1839, 4372, 2065, 3973

Offset: 0

Clark Kimberling, Mar 31 2012

nonn

For a guide to related sequences, see A210000.

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

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

Cf. A210000, A211053.

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 - 1], {n, 1, z1}]  (* A211054 *)

Offset corrected by Chai Wah Wu, Jan 04 2017

A211055 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant n+1.

Original entry on oeis.org

0, 1, 4, 9, 20, 23, 51, 55, 88, 79, 181, 121, 214, 222, 323, 227, 484, 299, 605, 488, 604, 457, 1102, 623, 902, 876, 1269, 769, 1754, 893, 1728, 1370, 1634, 1436, 2811, 1303, 2100, 1964, 3184, 1637, 3622, 1823, 3403, 3171, 3138

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..3000

Cf. A210000, A211053.

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 + 1], {n, 1, z1}]   (* A211055 *)

A211056 Number of 2 X 2 nonsingular matrices having all terms in {1,...,n}.

Original entry on oeis.org

0, 10, 66, 224, 576, 1210, 2290, 3936, 6352, 9722, 14322, 20304, 28080, 37834, 49922, 64704, 82624, 103898, 129170, 158640, 192944, 232554, 278050, 329680, 388368, 454522, 528770, 611680, 704192, 806490, 919890, 1044624, 1181680

Offset: 1

Clark Kimberling, Mar 31 2012

nonn

A211056(n) + A134506(n) = n^4.

For a guide to related sequences, see A210000.

Cf. A134506, A210000, A211059.

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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n^2] - c[n, 0], {n, 1, z1}]   (* A211059 *)
2*%     (* A211056 *)

A211057 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant in the closed interval [0,n].

Original entry on oeis.org

1, 10, 36, 91, 179, 330, 516, 802, 1150, 1615, 2119, 2873, 3595, 4558, 5653, 6967, 8245, 10020, 11642, 13846, 16053, 18524, 20944, 24393, 27405, 30924, 34637, 39035, 42961, 48396, 52906, 58687, 64326, 70457, 76722, 84824, 91318, 99045

Offset: 1

Clark Kimberling, Mar 31 2012

nonn

For a guide to related sequences, see A210000.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A210000.

g:= proc(n) local T,S,a,b,t,i;
  T:= Vector(n^2):
  for a from 1 to n do T[a^2]:= 1 od:
  for a from 1 to n-1 do for b from a+1 to n do
    T[a*b]:= T[a*b]+2
  od od;
  S:= Vector(n^2);
  S[1]:= T[1];
  for i from 2 to n^2 do S[i]:= S[i-1]+T[i] od;
  t:= T[1]*S[n+1];
  for i from 2 to n^2-n do
    t:= t + T[i]*(S[i+n]-S[i-1])
  od;
  t+1
end proc:
g(1):= 1:
map(g, [$1..40]); # Robert Israel, Sep 06 2024

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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n], {n, 1, z1}]

cp's OEIS Frontend

A210366 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant >=n.

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A210367 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant >= 2n.

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A210368 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant >= 3n.

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A210375 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 3.

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A210377 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 5.

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A210698 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant = 0 (mod 3).

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A211054 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant n-1.

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A211055 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant n+1.

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A211056 Number of 2 X 2 nonsingular matrices having all terms in {1,...,n}.