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

0,2

Comments

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

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

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)

Mathematica

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 *)

Crossrefs

Cf. A171503.

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

Sequence in context: A345332 A183023 A284246 * A134067 A350107 A024932

Adjacent sequences: A209997 A209998 A209999 * A210001 A210002 A210003

Keyword

nonn

Author

Clark Kimberling, Mar 16 2012

Extensions

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

Status

approved