A210369 -id:A210369 - cp's OEIS frontend

A210000 Number of unimodular 2 X 2 matrices having all terms in {0,1,...,n}.

0, 6, 14, 30, 46, 78, 94, 142, 174, 222, 254, 334, 366, 462, 510, 574, 638, 766, 814, 958, 1022, 1118, 1198, 1374, 1438, 1598, 1694, 1838, 1934, 2158, 2222, 2462, 2590, 2750, 2878, 3070, 3166, 3454, 3598, 3790, 3918, 4238, 4334, 4670, 4830

Offset: 0

Clark Kimberling, Mar 16 2012

nonn

a(n) is the number of 2 X 2 matrices having all terms in {0,1,...,n} and inverses with all terms integers.
Most sequences in the following guide count 2 X 2 matrices having all terms contained in the domain shown in column 2 and determinant d or permanent p or sum s of terms as indicated in column 3.
A059306 ... {0,1,...,n} ..... d=0
A171503 ... {0,1,...,n} ..... d=1
A210000 ... {0,1,...,n} .... |d|=1
A209973 ... {0,1,...,n} ..... d=2
A209975 ... {0,1,...,n} ..... d=3
A209976 ... {0,1,...,n} ..... d=4
A209977 ... {0,1,...,n} ..... d=5
A210282 ... {0,1,...,n} ..... d=n
A210283 ... {0,1,...,n} ..... d=n-1
A210284 ... {0,1,...,n} ..... d=n+1
A210285 ... {0,1,...,n} ..... d=floor(n/2)
A210286 ... {0,1,...,n} ..... d=trace
A280588 ... {0,1,...,n} ..... d=s
A106634 ... {0,1,...,n} ..... p=n
A210288 ... {0,1,...,n} ..... p=trace
A210289 ... {0,1,...,n} ..... p=(trace)^2
A280934 ... {0,1,...,n} ..... p=s
A210290 ... {0,1,...,n} ..... d>=0
A183761 ... {0,1,...,n} ..... d>0
A210291 ... {0,1,...,n} ..... d>n
A210366 ... {0,1,...,n} ..... d>=n
A210367 ... {0,1,...,n} ..... d>=2n
A210368 ... {0,1,...,n} ..... d>=3n
A210369 ... {0,1,...,n} ..... d is even
A210370 ... {0,1,...,n} ..... d is odd
A210371 ... {0,1,...,n} ..... d is even and >=0
A210372 ... {0,1,...,n} ..... d is even and >0
A210373 ... {0,1,...,n} ..... d is odd and >0
A210374 ... {0,1,...,n} ..... s=n+2
A210375 ... {0,1,...,n} ..... s=n+3
A210376 ... {0,1,...,n} ..... s=n+4
A210377 ... {0,1,...,n} ..... s=n+5
A210378 ... {0,1,...,n} ..... t is even
A210379 ... {0,1,...,n} ..... t is odd
A211031 ... {0,1,...,n} ..... d is in [-n,n]
A211032 ... {0,1,...,n} ..... d is in (-n,n)
A211033 ... {0,1,...,n} ..... d=0 (mod 3)
A211034 ... {0,1,...,n} ..... d=1 (mod 3)
A209974 = (A209973)/4
A134506 ... {1,2,...,n} ..... d=0
A196227 ... {1,2,...,n} ..... d=1
A209979 ... {1,2,...,n} .... |d|=1
A197168 ... {1,2,...,n} ..... d=2
A210001 ... {1,2,...,n} ..... d=3
A210002 ... {1,2,...,n} ..... d=4
A210027 ... {1,2,...,n} ..... d=5
A209978 = (A196227)/2
A209980 = (A197168)/2
A211053 ... {1,2,...,n} ..... d=n
A211054 ... {1,2,...,n} ..... d=n-1
A211055 ... {1,2,...,n} ..... d=n+1
A055507 ... {1,2,...,n} ..... p=n
A211057 ... {1,2,...,n} ..... d is in [0,n]
A211058 ... {1,2,...,n} ..... d>=0
A211059 ... {1,2,...,n} ..... d>0
A211060 ... {1,2,...,n} ..... d>n
A211061 ... {1,2,...,n} ..... d>=n
A211062 ... {1,2,...,n} ..... d>=2n
A211063 ... {1,2,...,n} ..... d>=3n
A211064 ... {1,2,...,n} ..... d is even
A211065 ... {1,2,...,n} ..... d is odd
A211066 ... {1,2,...,n} ..... d is even and >=0
A211067 ... {1,2,...,n} ..... d is even and >0
A211068 ... {1,2,...,n} ..... d is odd and >0
A209981 ... {-n,....,n} ..... d=0
A209982 ... {-n,....,n} ..... d=1
A209984 ... {-n,....,n} ..... d=2
A209986 ... {-n,....,n} ..... d=3
A209988 ... {-n,....,n} ..... d=4
A209990 ... {-n,....,n} ..... d=5
A211140 ... {-n,....,n} ..... d=n
A211141 ... {-n,....,n} ..... d=n-1
A211142 ... {-n,....,n} ..... d=n+1
A211143 ... {-n,....,n} ..... d=n^2
A211140 ... {-n,....,n} ..... p=n
A211145 ... {-n,....,n} ..... p=trace
A211146 ... {-n,....,n} ..... d in [0,n]
A211147 ... {-n,....,n} ..... d>=0
A211148 ... {-n,....,n} ..... d>0
A211149 ... {-n,....,n} ..... d<0 or d>0
A211150 ... {-n,....,n} ..... d>n
A211151 ... {-n,....,n} ..... d>=n
A211152 ... {-n,....,n} ..... d>=2n
A211153 ... {-n,....,n} ..... d>=3n
A211154 ... {-n,....,n} ..... d is even
A211155 ... {-n,....,n} ..... d is odd
A211156 ... {-n,....,n} ..... d is even and >=0
A211157 ... {-n,....,n} ..... d is even and >0
A211158 ... {-n,....,n} ..... d is odd and >0

			a(2)=6 counts these matrices (using reduced matrix notation):
(1,0,0,1), determinant = 1, inverse = (1,0,0,1)
(1,0,1,1), determinant = 1, inverse = (1,0,-1,1)
(1,1,0,1), determinant = 1, inverse = (1,-1,0,1)
(0,1,1,0), determinant = -1, inverse = (0,1,1,0)
(0,1,1,1), determinant = -1, inverse = (-1,1,1,0)
(1,1,1,0), determinant = -1, inverse = (0,1,1,-1)

Cf. A171503.

See also the very useful list of cross-references in the Comments section.

a = 0; b = n; z1 = 50;
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, 0, z1}]  (* A059306 *)
Table[c[n, 1], {n, 0, z1}]  (* A171503 *)
2 %                         (* A210000 *)
Table[c[n, 2], {n, 0, z1}]  (* A209973 *)
%/4                         (* A209974 *)
Table[c[n, 3], {n, 0, z1}]  (* A209975 *)
Table[c[n, 4], {n, 0, z1}]  (* A209976 *)
Table[c[n, 5], {n, 0, z1}]  (* A209977 *)

a(n) = 2*A171503(n).

A209982 added to list in comment by Chai Wah Wu, Nov 27 2016

A210370 Number of 2 X 2 matrices with all elements in {0,1,...,n} and odd determinant.

Original entry on oeis.org

0, 6, 16, 96, 168, 486, 720, 1536, 2080, 3750, 4800, 7776, 9576, 14406, 17248, 24576, 28800, 39366, 45360, 60000, 68200, 87846, 98736, 124416, 138528, 171366, 189280, 230496, 252840, 303750, 331200, 393216, 426496, 501126, 541008, 629856, 677160, 781926, 837520

Offset: 0

Clark Kimberling, Mar 20 2012

nonn

a(n) is also the number of 2 X 2 matrices with all elements in {0,1,...n} and odd permanent.

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, A210369.

a = 0; b = n; z1 = 28;
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, 0,  z1}] (* A210369 *)
Table[v[n], {n, 0, z1}](* A210370 *)

a(n)={2*((n+1)^2-ceil(n/2)^2)*ceil(n/2)^2} \\ Andrew Howroyd, Apr 28 2020

a(n) + A210369(n) = n^4.
From Colin Barker, Nov 28 2014: (Start)
a(n) = (3 - 3*(-1)^n - 12*(-1+(-1)^n)*n + (22-14*(-1)^n)*n^2 - 4*(-5+(-1)^n)*n^3 + 6*n^4)/16.
G.f.: -2*x*(3*x^5+17*x^4+16*x^3+28*x^2+5*x+3) / ((x-1)^5*(x+1)^4).
(End)
a(n) = 2*((n+1)^2 - ceiling(n/2)^2)*ceiling(n/2)^2. - Andrew Howroyd, Apr 28 2020

Terms a(29) and beyond from Andrew Howroyd, Apr 28 2020

A210371 Number of 2 X 2 matrices with all elements in {0,1,...,n} and nonnegative even determinant.

Original entry on oeis.org

1, 10, 48, 112, 285, 490, 968, 1448, 2465, 3410, 5280, 6904, 10021, 12610, 17400, 21312, 28321, 33866, 43704, 51336, 64661, 74898, 92416, 105680, 128297, 145234, 173712, 194928, 230333, 256410, 299776

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..10000

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]
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, 0, z1}] (* A210371 *)
Table[v[n], {n, 0, z1}] (* A210372 *)
Table[w[n], {n, 0, z1}] (* A210373 *)

a(n) = (A210369(n) + A059306(n))/2. - Chai Wah Wu, Nov 27 2016

A210372 Number of 2 X 2 matrices with all elements in {0,1,...,n} and positive even determinant.

Original entry on oeis.org

0, 0, 17, 48, 172, 320, 713, 1112, 2016, 2840, 4561, 6056, 8964, 11400, 15977, 19648, 26400, 31744, 41257, 48664, 61620, 71512, 88689, 101680, 123800, 140376, 168449, 189232, 224108, 249840, 292545

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..10000

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]
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, 0, z1}] (* A210371 *)
Table[v[n], {n, 0, z1}] (* A210372 *)
Table[w[n], {n, 0, z1}] (* A210373 *)

a(n) = (A210369(n) - A059306(n))/2. - Chai Wah Wu, Nov 27 2016

Offset corrected by Chai Wah Wu, Nov 27 2016

A277044 Number of 2 X 2 matrices with entries in {0,1,...,n} and even determinant with no entry repeated.

Original entry on oeis.org

0, 0, 0, 16, 96, 216, 600, 1008, 2064, 3040, 5280, 7200, 11280, 14616, 21336, 26656, 36960, 44928, 59904, 71280, 92160, 107800, 135960, 156816, 193776, 220896, 268320, 302848, 362544, 405720, 479640, 532800, 623040, 687616, 796416, 873936, 1003680, 1095768, 1248984, 1357360, 1536720, 1663200

Offset: 0

Indranil Ghosh, Dec 12 2016

a(n) mod 8 = 0.

Indranil Ghosh, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A210369 (where the entries can be repeated).

F(n,{r=0})={my(s=vector(2),v);forvec(y=vector(4,j,[0,n]),for(k=23*!!r,23,v=numtoperm(4,k);s[1+(y[v[1]]*y[v[4]]-y[v[3]]*y[v[2]])%2]++),2*!r);return(s)} \\ Use r=1 for A210369;
a(n)=F(n,0)[1]; \\ Also works for A210370 if F(n,1)[2] is used instead. - R. J. Cano, Dec 12 2016

a(n)=my(e=n\2+1,o=(n+1)\2); 24*binomial(o,4) + 16*binomial(e,2)*binomial(o,2) + 24*o*binomial(e,3) + 24*binomial(e,4) \\ Charles R Greathouse IV, Dec 12 2016

def t(n):
    s=0
    for a in range(0,n+1):
        for b in range(0,n+1):
            for c in range(0,n+1):
                for d in range(0,n+1):
                   if (a!=b  and a!=d and b!=d and c!=a and c!=b and c!=d):
                        if (a*d-b*c)%2==0:
                            s+=1
    return s
for i in range(0,201):
    print(f"{i} {t(i)}")

From Colin Barker and Charles R Greathouse IV, Dec 12 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>8.
a(n) = (5*n^4 - 8*n^3 + 4*n^2 - 16*n)/8 for n even.
a(n) = (5*n^4 - 12*n^3 + 2*n^2 + 12*n - 7)/8 for n odd.
G.f.: 8*x^3*(2 + 10*x + 7*x^2 + 8*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^4).
(End)

cp's OEIS Frontend

A210000 Number of unimodular 2 X 2 matrices having all terms in {0,1,...,n}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A210370 Number of 2 X 2 matrices with all elements in {0,1,...,n} and odd determinant.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A210371 Number of 2 X 2 matrices with all elements in {0,1,...,n} and nonnegative even determinant.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A210372 Number of 2 X 2 matrices with all elements in {0,1,...,n} and positive even determinant.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A277044 Number of 2 X 2 matrices with entries in {0,1,...,n} and even determinant with no entry repeated.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Python

Formula