A210000 - OEIS

A210000

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

101

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

(list; graph; refs; listen; history; text; internal format)

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