A210000 -id:A210000 - cp's OEIS frontend

A210289 Number of 2 X 2 matrices with all elements in {0,1,...,n} and permanent = (trace)^2.

1, 5, 9, 15, 25, 29, 35, 53, 63, 81, 85, 89, 107, 141, 159, 165, 193, 197, 215, 261, 271, 323, 327, 331, 349, 389, 423, 461, 529, 533, 539, 617, 645, 651, 655, 673, 727, 817, 863, 959, 969, 973, 1025, 1131, 1141, 1159

Offset: 0

Clark Kimberling, Mar 19 2012

nonn

See A210000 for a guide to related sequences.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A210000.

a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w^2 + z^2 + 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, 0], {n, 0, z1}]  (* A210289 *)

A210376 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 4.

Original entry on oeis.org

0, 0, 10, 40, 85, 140, 206, 284, 375, 480, 600, 736, 889, 1060, 1250, 1460, 1691, 1944, 2220, 2520, 2845, 3196, 3574, 3980, 4415, 4880, 5376, 5904, 6465, 7060, 7690, 8356, 9059, 9800, 10580, 11400, 12261, 13164, 14110, 15100, 16135, 17216

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 - 4.

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 + 4], {n, 0, z1}]    (* A210376 *)
Table[c[n, 3 n - 4], {n, 0, z1}]  (* A210376 *)

Conjectures from Colin Barker, Dec 07 2017: (Start)
G.f.: x^2*(10 - 15*x^2 + 6*x^4) / (1 - x)^4.
a(n) = (-270 + 107*n + 18*n^2 + n^3) / 6 for n>2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A211058 Number of 2 X 2 matrices having all terms in {1,...,n} and nonnegative determinant.

Original entry on oeis.org

1, 11, 48, 144, 337, 691, 1256, 2128, 3385, 5139, 7480, 10584, 14521, 19499, 25664, 33184, 42209, 53027, 65736, 80680, 98009, 117979, 140816, 166936, 196441, 229715, 267056, 308816, 355185, 406755, 463576, 526264, 595081, 670419

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

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], {n, 1, z1}]   (* A211058 *)

A211060 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >n.

Original entry on oeis.org

0, 1, 12, 53, 158, 361, 740, 1326, 2235, 3524, 5361, 7711, 10926, 14941, 20011, 26217, 33964, 43007, 54094, 66834, 81956, 99455, 119872, 142543, 169036, 198791, 232419, 269781, 312224, 358359, 410670, 467577, 530755, 599962, 676006

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

A211061 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >= n.

Original entry on oeis.org

0, 3, 19, 69, 181, 411, 785, 1419, 2334, 3674, 5478, 7994, 11093, 15249, 20347, 26660, 34253, 43661, 54463, 67637, 82614, 100217, 120415, 143935, 169815, 199883, 233505, 271344, 313103, 360519, 411681, 469615, 532407, 601850, 677764

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 = 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] - c1[n, n - 1], {n, 1, z1}]

A211145 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and permanent=trace.

Original entry on oeis.org

1, 19, 60, 116, 196, 292, 404, 548, 708, 868, 1060, 1284, 1524, 1796, 2052, 2292, 2612, 2980, 3348, 3748, 4116, 4452, 4884, 5380, 5844, 6324, 6820, 7300, 7860, 8468, 9060, 9732, 10388, 10964, 11572, 12180, 12852, 13668, 14436, 15092

Offset: 0

Clark Kimberling, Apr 03 2012

nonn

For a guide to related sequences, see A210000.

David Radcliffe, Table of n, a(n) for n = 0..2000

Cf. A210000.

a = -n; b = n; z1 = 40;
t[n_] := t[n] = Flatten[Table[w + z - 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, 0], {n, 0, z1}]   (* A211145 *)
(1/4) Table[c[n, n], {n, 2, z1}] (* integers *)

A211148 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive determinant.

Original entry on oeis.org

24, 248, 1056, 3008, 6904, 13624, 24448, 40576, 63640, 95288, 137632, 192384, 262392, 349688, 457088, 587520, 744344, 930104, 1149152, 1404160, 1699640, 2039544, 2428352, 2869312, 3368472, 3929912, 4558688, 5259712, 6039480

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]
Table[c[n, 0], {n, 1, z1}]   (* A209981 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 1, m}]
t = Table[c1[n, 2*n^2], {n, 1, z1}]   (* A211148 *)
2 t   (* A211149 *)
t/8   (* integers *)

A211149 Number of 2 X 2 nonsingular matrices having all terms in {-n,...,0,...,n}.

Original entry on oeis.org

48, 496, 2112, 6016, 13808, 27248, 48896, 81152, 127280, 190576, 275264, 384768, 524784, 699376, 914176, 1175040, 1488688, 1860208, 2298304, 2808320, 3399280, 4079088, 4856704, 5738624, 6736944, 7859824, 9117376, 10519424

Offset: 1

Clark Kimberling, Apr 04 2012

nonn

A211149(n) + A209981(n) = (2n+1)^4 for n>0.
It appears that 16 divides A211149(n).
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]
Table[c[n, 0], {n, 1, z1}]   (* A209981 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 1, m}]
t = Table[c1[n, 2*n^2], {n, 1, z1}]   (* A211148 *)
2 t   (* A211149 *)
t/8   (* integers *)

A211154 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and even determinant.

Original entry on oeis.org

1, 41, 457, 1345, 4481, 8521, 18985, 30017, 54721, 78121, 126281, 168961, 252097, 322505, 454441, 562561, 759425, 916777, 1197001, 1416641, 1800961, 2097481, 2608937, 2998465, 3662401, 4162601, 5006665, 5636737, 6690881, 7471561, 8768041, 9721601, 11294977, 12445225, 14332361, 15704641

Offset: 0

Clark Kimberling, Apr 05 2012

nonn

For a guide to related sequences, see A210000.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A210000, A211155.

seq((2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n), n=1..20); # Mark van Hoeij, May 13 2013

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 *)

a(n)=(2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n); \\ Joerg Arndt, May 14 2013

a(n) + A211155(n) = (2n+1)^4.
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*(-x^8 - 36*x^6 - 416*x^5 - 734*x^4 - 1472*x^3 - 724*x^2 - 416*x - 41)/((x - 1)^5*(x + 1)^4). (End)

More terms from Joerg Arndt, May 14 2013

a(0)=1 prepended by Andrew Howroyd, May 05 2020

A281315 Number of 2 X 2 matrices with all elements in {0,...,n} and prime determinant.

Original entry on oeis.org

0, 0, 13, 46, 83, 191, 272, 509, 687, 1010, 1291, 2019, 2364, 3468, 4132, 5079, 6072, 8298, 9234, 12189, 13621, 15984, 18095, 22965, 24886, 29942, 33248, 38385, 42073, 51053, 53882, 64609, 70619, 78663, 85424, 96024, 101521, 118804, 127940, 140598, 149375, 172123, 179424, 205334, 218216

Offset: 0

Indranil Ghosh, Jan 20 2017

nonn

			For n = 3, a few of the possible matrices are [1,0;3,3], [1,1;0,2], [1,1;0,3], [1,1;1,3], [1,2;0,2], [1,2;0,3], [1,3;0,2], [1,3;0,3], [2,0;0,1], [2,0;1,1], [2,0;2,1], [2,0;3,1], [2,1;0,1], [2,1;1,2], [2,1;1,3], [3,1;3,2], [3,2;0,1], [3,2;1,3], [3,2;2,2], [3,2;2,3], ... There are 46 possibilities.
Here each of the matrices M is defined as M = [a,b;c,d], where a= M[1][1], b = M[1][2], c = M[2][1] and d = M[2][2]. So, a(3) = 46.

Chai Wah Wu, Table of n, a(n) for n = 0..1000 (terms 0..186 from Indranil Ghosh)

Cf. A210000.

from sympy import isprime
def t(n):
    s=0
    for a in range(n+1):
        for d in range(n+1):
            ad = a * d
            for c in range(n+1):
                for b in range(n+1):
                    if isprime(ad-b*c):
                        s+=1
    return s
for i in range(187):
    print(str(i)+" "+str(t(i)))

def A281315(n):
    T = Tuples([i for i in range(n+1)], 4); i = 0
    for t in T: i += is_prime(t[0]*t[3]-t[1]*t[2])
    return i
[A281315(n) for n in range(20)] # Peter Luschny, Jul 23 2017

cp's OEIS Frontend

A210289 Number of 2 X 2 matrices with all elements in {0,1,...,n} and permanent = (trace)^2.

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A210376 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 4.

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Formula

A211058 Number of 2 X 2 matrices having all terms in {1,...,n} and nonnegative determinant.

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A211060 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >n.

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A211061 Number of 2 X 2 matrices having all terms in {1,...,n} and determinant >= n.

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A211145 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and permanent=trace.

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A211148 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive determinant.

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A211149 Number of 2 X 2 nonsingular matrices having all terms in {-n,...,0,...,n}.

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A211154 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and even determinant.

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