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 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

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Last modified May 29 07:06 EDT 2024. Contains 372926 sequences. (Running on oeis4.)