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0, 0, 1, 4, 7, 14, 17, 28, 35, 46, 53, 72, 79, 102, 113, 128, 143, 174, 185, 220, 235, 258, 277, 320, 335, 374, 397, 432, 455, 510, 525, 584, 615, 654, 685, 732, 755, 826, 861, 908, 939, 1018, 1041, 1124, 1163, 1210, 1253, 1344, 1375, 1458, 1497
COMMENTS
See A210000 for a guide to related sequences.
MAPLE
a:= proc(n) option remember; `if`(n<2, 0,
a(n-1)-1 + 2*numtheory[phi](n))
end:
MATHEMATICA
a = 1; 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}] (* A134506 *) Table[c[n, 1], {n, 0, z1}] (* A196227 *) Table[2 c[n, 1], {n, 0, z1}](* A209979 *) Table[c[n, 2], {n, 0, z1}] (* A197168 *) Table[c[n, 3], {n, 0, z1}] (* A210001 *) Table[c[n, 4], {n, 0, z1}] (* A210002 *) Table[c[n, 5], {n, 0, z1}] (* A210027 *)
Number of unimodular 2 X 2 matrices having all terms in {0,1,...,n}.
+10
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
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.
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
A210288 ... {0,1,...,n} ..... p=trace
A210289 ... {0,1,...,n} ..... p=(trace)^2
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)
A211054 ... {1,2,...,n} ..... d=n-1
A211055 ... {1,2,...,n} ..... d=n+1
A211057 ... {1,2,...,n} ..... d is in [0,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
A211141 ... {-n,....,n} ..... d=n-1
A211142 ... {-n,....,n} ..... d=n+1
A211143 ... {-n,....,n} ..... d=n^2
A211145 ... {-n,....,n} ..... p=trace
A211146 ... {-n,....,n} ..... d in [0,n]
A211149 ... {-n,....,n} ..... d<0 or d>0
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
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 *) Table[c[n, 2], {n, 0, z1}] (* A209973 *) 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
See also the very useful list of cross-references in the Comments section.
Number of 2 X 2 matrices having all elements in {-n,...,n} and determinant 1.
+10
4
0, 20, 52, 116, 180, 308, 372, 564, 692, 884, 1012, 1332, 1460, 1844, 2036, 2292, 2548, 3060, 3252, 3828, 4084, 4468, 4788, 5492, 5748, 6388, 6772, 7348, 7732, 8628, 8884, 9844, 10356, 10996, 11508, 12276, 12660, 13812, 14388, 15156
COMMENTS
See A210000 for a guide to related sequences.
FORMULA
a(n) = -12 + 32*Sum_{k=1..n} phi(k) for n > 0. (End)
MATHEMATICA
(See the Mathematica section at A209981.)
PROG
(PARI) a(n)=if(n<1, 0, 32*sum(k=1, n, eulerphi(k)) - 12) \\ Andrew Howroyd, May 05 2020
Number of unimodular 2 X 2 matrices having all elements in {1,2,...,n}.
+10
3
0, 0, 4, 16, 28, 56, 68, 112, 140, 184, 212, 288, 316, 408, 452, 512, 572, 696, 740, 880, 940, 1032, 1108, 1280, 1340, 1496, 1588, 1728, 1820, 2040, 2100, 2336, 2460, 2616, 2740, 2928, 3020, 3304, 3444, 3632, 3756, 4072, 4164, 4496, 4652, 4840
COMMENTS
Equivalently, the number of 2 X 2 matrices having all elements in {1,2,...,n} and having an inverse whose elements are all integers.
See A210000 for a guide to related sequences.
MATHEMATICA
(See the Mathematica section at A210000.)
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