cp's OEIS Frontend

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

%I A210000 #33 Jan 12 2017 15:58:59
%S A210000 0,6,14,30,46,78,94,142,174,222,254,334,366,462,510,574,638,766,814,
%T A210000 958,1022,1118,1198,1374,1438,1598,1694,1838,1934,2158,2222,2462,2590,
%U A210000 2750,2878,3070,3166,3454,3598,3790,3918,4238,4334,4670,4830
%N A210000 Number of unimodular 2 X 2 matrices having all terms in {0,1,...,n}.
%C A210000 a(n) is the number of 2 X 2 matrices having all terms in {0,1,...,n} and inverses with all terms integers.
%C A210000 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 A210000 A059306 ... {0,1,...,n} ..... d=0
%C A210000 A171503 ... {0,1,...,n} ..... d=1
%C A210000 A210000 ... {0,1,...,n} .... |d|=1
%C A210000 A209973 ... {0,1,...,n} ..... d=2
%C A210000 A209975 ... {0,1,...,n} ..... d=3
%C A210000 A209976 ... {0,1,...,n} ..... d=4
%C A210000 A209977 ... {0,1,...,n} ..... d=5
%C A210000 A210282 ... {0,1,...,n} ..... d=n
%C A210000 A210283 ... {0,1,...,n} ..... d=n-1
%C A210000 A210284 ... {0,1,...,n} ..... d=n+1
%C A210000 A210285 ... {0,1,...,n} ..... d=floor(n/2)
%C A210000 A210286 ... {0,1,...,n} ..... d=trace
%C A210000 A280588 ... {0,1,...,n} ..... d=s
%C A210000 A106634 ... {0,1,...,n} ..... p=n
%C A210000 A210288 ... {0,1,...,n} ..... p=trace
%C A210000 A210289 ... {0,1,...,n} ..... p=(trace)^2
%C A210000 A280934 ... {0,1,...,n} ..... p=s
%C A210000 A210290 ... {0,1,...,n} ..... d>=0
%C A210000 A183761 ... {0,1,...,n} ..... d>0
%C A210000 A210291 ... {0,1,...,n} ..... d>n
%C A210000 A210366 ... {0,1,...,n} ..... d>=n
%C A210000 A210367 ... {0,1,...,n} ..... d>=2n
%C A210000 A210368 ... {0,1,...,n} ..... d>=3n
%C A210000 A210369 ... {0,1,...,n} ..... d is even
%C A210000 A210370 ... {0,1,...,n} ..... d is odd
%C A210000 A210371 ... {0,1,...,n} ..... d is even and >=0
%C A210000 A210372 ... {0,1,...,n} ..... d is even and >0
%C A210000 A210373 ... {0,1,...,n} ..... d is odd and >0
%C A210000 A210374 ... {0,1,...,n} ..... s=n+2
%C A210000 A210375 ... {0,1,...,n} ..... s=n+3
%C A210000 A210376 ... {0,1,...,n} ..... s=n+4
%C A210000 A210377 ... {0,1,...,n} ..... s=n+5
%C A210000 A210378 ... {0,1,...,n} ..... t is even
%C A210000 A210379 ... {0,1,...,n} ..... t is odd
%C A210000 A211031 ... {0,1,...,n} ..... d is in [-n,n]
%C A210000 A211032 ... {0,1,...,n} ..... d is in (-n,n)
%C A210000 A211033 ... {0,1,...,n} ..... d=0 (mod 3)
%C A210000 A211034 ... {0,1,...,n} ..... d=1 (mod 3)
%C A210000 A209974 = (A209973)/4
%C A210000 A134506 ... {1,2,...,n} ..... d=0
%C A210000 A196227 ... {1,2,...,n} ..... d=1
%C A210000 A209979 ... {1,2,...,n} .... |d|=1
%C A210000 A197168 ... {1,2,...,n} ..... d=2
%C A210000 A210001 ... {1,2,...,n} ..... d=3
%C A210000 A210002 ... {1,2,...,n} ..... d=4
%C A210000 A210027 ... {1,2,...,n} ..... d=5
%C A210000 A209978 = (A196227)/2
%C A210000 A209980 = (A197168)/2
%C A210000 A211053 ... {1,2,...,n} ..... d=n
%C A210000 A211054 ... {1,2,...,n} ..... d=n-1
%C A210000 A211055 ... {1,2,...,n} ..... d=n+1
%C A210000 A055507 ... {1,2,...,n} ..... p=n
%C A210000 A211057 ... {1,2,...,n} ..... d is in [0,n]
%C A210000 A211058 ... {1,2,...,n} ..... d>=0
%C A210000 A211059 ... {1,2,...,n} ..... d>0
%C A210000 A211060 ... {1,2,...,n} ..... d>n
%C A210000 A211061 ... {1,2,...,n} ..... d>=n
%C A210000 A211062 ... {1,2,...,n} ..... d>=2n
%C A210000 A211063 ... {1,2,...,n} ..... d>=3n
%C A210000 A211064 ... {1,2,...,n} ..... d is even
%C A210000 A211065 ... {1,2,...,n} ..... d is odd
%C A210000 A211066 ... {1,2,...,n} ..... d is even and >=0
%C A210000 A211067 ... {1,2,...,n} ..... d is even and >0
%C A210000 A211068 ... {1,2,...,n} ..... d is odd and >0
%C A210000 A209981 ... {-n,....,n} ..... d=0
%C A210000 A209982 ... {-n,....,n} ..... d=1
%C A210000 A209984 ... {-n,....,n} ..... d=2
%C A210000 A209986 ... {-n,....,n} ..... d=3
%C A210000 A209988 ... {-n,....,n} ..... d=4
%C A210000 A209990 ... {-n,....,n} ..... d=5
%C A210000 A211140 ... {-n,....,n} ..... d=n
%C A210000 A211141 ... {-n,....,n} ..... d=n-1
%C A210000 A211142 ... {-n,....,n} ..... d=n+1
%C A210000 A211143 ... {-n,....,n} ..... d=n^2
%C A210000 A211140 ... {-n,....,n} ..... p=n
%C A210000 A211145 ... {-n,....,n} ..... p=trace
%C A210000 A211146 ... {-n,....,n} ..... d in [0,n]
%C A210000 A211147 ... {-n,....,n} ..... d>=0
%C A210000 A211148 ... {-n,....,n} ..... d>0
%C A210000 A211149 ... {-n,....,n} ..... d<0 or d>0
%C A210000 A211150 ... {-n,....,n} ..... d>n
%C A210000 A211151 ... {-n,....,n} ..... d>=n
%C A210000 A211152 ... {-n,....,n} ..... d>=2n
%C A210000 A211153 ... {-n,....,n} ..... d>=3n
%C A210000 A211154 ... {-n,....,n} ..... d is even
%C A210000 A211155 ... {-n,....,n} ..... d is odd
%C A210000 A211156 ... {-n,....,n} ..... d is even and >=0
%C A210000 A211157 ... {-n,....,n} ..... d is even and >0
%C A210000 A211158 ... {-n,....,n} ..... d is odd and >0
%F A210000 a(n) = 2*A171503(n).
%e A210000 a(2)=6 counts these matrices (using reduced matrix notation):
%e A210000 (1,0,0,1), determinant = 1, inverse = (1,0,0,1)
%e A210000 (1,0,1,1), determinant = 1, inverse = (1,0,-1,1)
%e A210000 (1,1,0,1), determinant = 1, inverse = (1,-1,0,1)
%e A210000 (0,1,1,0), determinant = -1, inverse = (0,1,1,0)
%e A210000 (0,1,1,1), determinant = -1, inverse = (-1,1,1,0)
%e A210000 (1,1,1,0), determinant = -1, inverse = (0,1,1,-1)
%t A210000 a = 0; b = n; z1 = 50;
%t A210000 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210000 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210000 Table[c[n, 0], {n, 0, z1}]  (* A059306 *)
%t A210000 Table[c[n, 1], {n, 0, z1}]  (* A171503 *)
%t A210000 2 %                         (* A210000 *)
%t A210000 Table[c[n, 2], {n, 0, z1}]  (* A209973 *)
%t A210000 %/4                         (* A209974 *)
%t A210000 Table[c[n, 3], {n, 0, z1}]  (* A209975 *)
%t A210000 Table[c[n, 4], {n, 0, z1}]  (* A209976 *)
%t A210000 Table[c[n, 5], {n, 0, z1}]  (* A209977 *)
%Y A210000 Cf. A171503.
%Y A210000 See also the very useful list of cross-references in the Comments section.
%K A210000 nonn
%O A210000 0,2
%A A210000 _Clark Kimberling_, Mar 16 2012
%E A210000 A209982 added to list in comment by _Chai Wah Wu_, Nov 27 2016