A210000 - OEIS

%I #33 Jan 12 2017 15:58:59

%S 0,6,14,30,46,78,94,142,174,222,254,334,366,462,510,574,638,766,814,

%T 958,1022,1118,1198,1374,1438,1598,1694,1838,1934,2158,2222,2462,2590,

%U 2750,2878,3070,3166,3454,3598,3790,3918,4238,4334,4670,4830

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

%C a(n) is the number of 2 X 2 matrices having all terms in {0,1,...,n} and inverses with all terms integers.

%C 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.

%C A059306 ... {0,1,...,n} ..... d=0

%C A171503 ... {0,1,...,n} ..... d=1

%C A210000 ... {0,1,...,n} .... |d|=1

%C A209973 ... {0,1,...,n} ..... d=2

%C A209975 ... {0,1,...,n} ..... d=3

%C A209976 ... {0,1,...,n} ..... d=4

%C A209977 ... {0,1,...,n} ..... d=5

%C A210282 ... {0,1,...,n} ..... d=n

%C A210283 ... {0,1,...,n} ..... d=n-1

%C A210284 ... {0,1,...,n} ..... d=n+1

%C A210285 ... {0,1,...,n} ..... d=floor(n/2)

%C A210286 ... {0,1,...,n} ..... d=trace

%C A280588 ... {0,1,...,n} ..... d=s

%C A106634 ... {0,1,...,n} ..... p=n

%C A210288 ... {0,1,...,n} ..... p=trace

%C A210289 ... {0,1,...,n} ..... p=(trace)^2

%C A280934 ... {0,1,...,n} ..... p=s

%C A210290 ... {0,1,...,n} ..... d>=0

%C A183761 ... {0,1,...,n} ..... d>0

%C A210291 ... {0,1,...,n} ..... d>n

%C A210366 ... {0,1,...,n} ..... d>=n

%C A210367 ... {0,1,...,n} ..... d>=2n

%C A210368 ... {0,1,...,n} ..... d>=3n

%C A210369 ... {0,1,...,n} ..... d is even

%C A210370 ... {0,1,...,n} ..... d is odd

%C A210371 ... {0,1,...,n} ..... d is even and >=0

%C A210372 ... {0,1,...,n} ..... d is even and >0

%C A210373 ... {0,1,...,n} ..... d is odd and >0

%C A210374 ... {0,1,...,n} ..... s=n+2

%C A210375 ... {0,1,...,n} ..... s=n+3

%C A210376 ... {0,1,...,n} ..... s=n+4

%C A210377 ... {0,1,...,n} ..... s=n+5

%C A210378 ... {0,1,...,n} ..... t is even

%C A210379 ... {0,1,...,n} ..... t is odd

%C A211031 ... {0,1,...,n} ..... d is in [-n,n]

%C A211032 ... {0,1,...,n} ..... d is in (-n,n)

%C A211033 ... {0,1,...,n} ..... d=0 (mod 3)

%C A211034 ... {0,1,...,n} ..... d=1 (mod 3)

%C A209974 = (A209973)/4

%C A134506 ... {1,2,...,n} ..... d=0

%C A196227 ... {1,2,...,n} ..... d=1

%C A209979 ... {1,2,...,n} .... |d|=1

%C A197168 ... {1,2,...,n} ..... d=2

%C A210001 ... {1,2,...,n} ..... d=3

%C A210002 ... {1,2,...,n} ..... d=4

%C A210027 ... {1,2,...,n} ..... d=5

%C A209978 = (A196227)/2

%C A209980 = (A197168)/2

%C A211053 ... {1,2,...,n} ..... d=n

%C A211054 ... {1,2,...,n} ..... d=n-1

%C A211055 ... {1,2,...,n} ..... d=n+1

%C A055507 ... {1,2,...,n} ..... p=n

%C A211057 ... {1,2,...,n} ..... d is in [0,n]

%C A211058 ... {1,2,...,n} ..... d>=0

%C A211059 ... {1,2,...,n} ..... d>0

%C A211060 ... {1,2,...,n} ..... d>n

%C A211061 ... {1,2,...,n} ..... d>=n

%C A211062 ... {1,2,...,n} ..... d>=2n

%C A211063 ... {1,2,...,n} ..... d>=3n

%C A211064 ... {1,2,...,n} ..... d is even

%C A211065 ... {1,2,...,n} ..... d is odd

%C A211066 ... {1,2,...,n} ..... d is even and >=0

%C A211067 ... {1,2,...,n} ..... d is even and >0

%C A211068 ... {1,2,...,n} ..... d is odd and >0

%C A209981 ... {-n,....,n} ..... d=0

%C A209982 ... {-n,....,n} ..... d=1

%C A209984 ... {-n,....,n} ..... d=2

%C A209986 ... {-n,....,n} ..... d=3

%C A209988 ... {-n,....,n} ..... d=4

%C A209990 ... {-n,....,n} ..... d=5

%C A211140 ... {-n,....,n} ..... d=n

%C A211141 ... {-n,....,n} ..... d=n-1

%C A211142 ... {-n,....,n} ..... d=n+1

%C A211143 ... {-n,....,n} ..... d=n^2

%C A211140 ... {-n,....,n} ..... p=n

%C A211145 ... {-n,....,n} ..... p=trace

%C A211146 ... {-n,....,n} ..... d in [0,n]

%C A211147 ... {-n,....,n} ..... d>=0

%C A211148 ... {-n,....,n} ..... d>0

%C A211149 ... {-n,....,n} ..... d<0 or d>0

%C A211150 ... {-n,....,n} ..... d>n

%C A211151 ... {-n,....,n} ..... d>=n

%C A211152 ... {-n,....,n} ..... d>=2n

%C A211153 ... {-n,....,n} ..... d>=3n

%C A211154 ... {-n,....,n} ..... d is even

%C A211155 ... {-n,....,n} ..... d is odd

%C A211156 ... {-n,....,n} ..... d is even and >=0

%C A211157 ... {-n,....,n} ..... d is even and >0

%C A211158 ... {-n,....,n} ..... d is odd and >0

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

%e a(2)=6 counts these matrices (using reduced matrix notation):

%e (1,0,0,1), determinant = 1, inverse = (1,0,0,1)

%e (1,0,1,1), determinant = 1, inverse = (1,0,-1,1)

%e (1,1,0,1), determinant = 1, inverse = (1,-1,0,1)

%e (0,1,1,0), determinant = -1, inverse = (0,1,1,0)

%e (0,1,1,1), determinant = -1, inverse = (-1,1,1,0)

%e (1,1,1,0), determinant = -1, inverse = (0,1,1,-1)

%t a = 0; b = n; z1 = 50;

%t t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]

%t c[n_, k_] := c[n, k] = Count[t[n], k]

%t Table[c[n, 0], {n, 0, z1}]  (* A059306 *)

%t Table[c[n, 1], {n, 0, z1}]  (* A171503 *)

%t 2 %                         (* A210000 *)

%t Table[c[n, 2], {n, 0, z1}]  (* A209973 *)

%t %/4                         (* A209974 *)

%t Table[c[n, 3], {n, 0, z1}]  (* A209975 *)

%t Table[c[n, 4], {n, 0, z1}]  (* A209976 *)

%t Table[c[n, 5], {n, 0, z1}]  (* A209977 *)

%Y Cf. A171503.

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

%K nonn

%O 0,2

%A _Clark Kimberling_, Mar 16 2012

%E A209982 added to list in comment by _Chai Wah Wu_, Nov 27 2016