A059306 -id:A059306 - cp's OEIS frontend

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

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

Offset: 0

Clark Kimberling, Mar 16 2012

nonn

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

			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)

Cf. A171503.

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

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 *)

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

A209982 added to list in comment by Chai Wah Wu, Nov 27 2016

A062801 Number of 2 X 2 non-singular integer matrices with entries from {0,...,n}.

Original entry on oeis.org

0, 6, 50, 192, 512, 1126, 2146, 3760, 6112, 9430, 13922, 19888, 27504, 37206, 49202, 63872, 81600, 102854, 127874, 157328, 191440, 230870, 276114, 327776, 386128, 452118, 526178, 608960, 701056, 803430, 916290, 1040976, 1177744, 1327606

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 19 2001

All these numbers are even because there are the same number of matrices with positive and negative determinants. [T. D. Noe, Oct 08 2008]

This sequence a(n) and A059306 give together all the 2 X 2 matrices with entries from {0, ..., n}, i.e. a(n) + A059306(n) = (n+1)^4.

Table[cnt=0; Do[If[a*d-b*c != 0, cnt++ ], {a,0,n}, {b,0,n}, {c,0,n}, {d,0,n}]; cnt, {n,0,10}] (* T. D. Noe, Oct 08 2008 *)

More terms from T. D. Noe, Oct 08 2008

A134506 Number of 2 X 2 singular integer matrices with elements from {1,...,n}.

Original entry on oeis.org

0, 1, 6, 15, 32, 49, 86, 111, 160, 209, 278, 319, 432, 481, 582, 703, 832, 897, 1078, 1151, 1360, 1537, 1702, 1791, 2096, 2257, 2454, 2671, 2976, 3089, 3510, 3631, 3952, 4241, 4502, 4831, 5360, 5505, 5798, 6143, 6704, 6865, 7478, 7647, 8144, 8721, 9078, 9263

Offset: 0

Graziano Aglietti (mg5055(AT)mclink.it), Jan 20 2008, Feb 04 2008

a(2k) is even. a(4k+i) = i (mod 4), for i = 0, 1, 2, 3. - Aldo González Lorenzo, Oct 14 2011

Charles R Greathouse IV and Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms for n = 1..1000 from Charles R Greathouse IV)
Sanying Shi, On the equation n1n2 = n3n4 and mean value of character sums, Journal of Number Theory, Volume 128, Issue 2, February 2008, Pages 313-321.

Cf. A059306 (similar but with elements from {0, ..., n}).

a = {}; For[n = 2, n < 50, n++, s = 0; For[j = 1, j < n + 1, j++, For[c = 1, c < n + 1, c++, s = s + Length[Select[Divisors[c*j], # < n + 1 && c*j/# < n + 1 &]]]]; AppendTo[a, s]]; a (* Stefan Steinerberger, Feb 06 2008 *)

a(n) = {my(nnb = 0); for (i=1, n, for (j=1, n, pij = i*j; for (k=1, n, for (l=1, n, if (pij == k*l, nnb++););););); nnb;} \\ Michel Marcus, Feb 03 2016

a(n)=sum(i=1,n,sum(j=1,n,my(ij=i*j);sumdiv(ij,k, k<=n && ij/k<=n))) \\ Charles R Greathouse IV, Feb 03 2016

a(n)=2*sum(i=2,n,sum(j=1,i-1,my(ij=i*j);sumdiv(ij,k, k<=n && ij/k<=n))) + sum(i=1,n,my(i2=i^2);sumdiv(i2,k, k<=n && i2/k<=n)) \\ Charles R Greathouse IV, Feb 03 2016

Shi proves that a(n) = kn^2 log n + cn^2 + O(n^e) where k = 12/Pi^2, e > 547/416 = 1.3149..., and c is a complicated constant given in the paper (see p. 320 and pp. 314-315). - Charles R Greathouse IV, Feb 03 2016

a(n) = A059306(n) - (2n+1)^2. - Chai Wah Wu, Nov 28 2016

More terms from Stefan Steinerberger, Feb 06 2008

a(0) added by Chai Wah Wu, Nov 28 2016

A187521 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with the array of 2X2 subblock determinants antisymmetric and no off-diagonal 2X2 subblock determinant zero.

Original entry on oeis.org

10, 31, 6, 64, 98, 2, 113, 450, 354, 0, 170, 1590, 4200, 1780, 0, 255, 3426, 34776, 73764, 12008, 0, 336, 8546, 107990, 1655170, 2048596, 157694, 0, 449, 13992, 475370, 8174314, 144075566, 129034386, 2808560, 0, 570, 26158, 978568, 70304076

Offset: 1

R. H. Hardin Mar 10 2011

Table starts
.10......31........64.......113........170......255....336...449.570
..6......98.......450......1590.......3426.....8546..13992.26158
..2.....354......4200.....34776.....107990...475370.978568
..0....1780.....73764...1655170....8174314.70304076
..0...12008...2048596.144075566.1201064094
..0..157694.129034386
..0.2808560
..0

			Some k=3 solutions for 5X5
..0..0..2..0..2....2..3..2..1..1....2..1..2..3..3....1..1..0..3..1
..2..1..1..2..0....2..3..3..3..2....2..1..1..1..2....1..1..1..0..2
..2..2..2..2..1....3..3..3..1..2....1..1..1..2..1....2..1..1..1..2
..2..0..1..1..1....2..1..3..1..1....1..2..1..2..2....1..2..1..1..1
..0..2..3..2..2....3..2..2..1..1....2..1..2..2..2....3..0..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..41

Row 1 is A059306

A187705 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock off diagonal and antidiagonal nonsingular and the array of 2X2 subblock determinants antisymmetric about the diagonal and antidiagonal.

Original entry on oeis.org

10, 31, 128, 64, 1489, 2, 113, 8272, 72, 34, 170, 31461, 322, 5030, 0, 255, 93476, 1718, 81456, 36, 2, 336, 235773, 2904, 875696, 530, 7410, 0, 449, 524284, 9834, 4225652, 4612, 230446, 474, 0, 570, 1062857, 13324, 22532156, 12898

Offset: 1

R. H. Hardin Mar 12 2011

Table starts
..10...31.....64....113.....170......255....336.....449.570
.128.1489...8272..31461...93476...235773.524284.1062857
...2...72....322...1718....2904.....9834..13324
..34.5030..81456.875696.4225652.22532156
...0...36....530...4612...12898
...2.7410.230446
...0..474
...0

			Some k=3 solutions for 5X5
..0..0..1..2..2....0..0..1..2..2....0..0..1..0..0....0..3..2..3..2
..2..2..0..2..2....3..3..0..1..1....1..1..0..1..2....0..3..3..3..2
..2..3..0..3..2....1..2..0..2..1....1..2..0..1..1....1..0..0..0..1
..2..2..0..2..2....1..1..0..3..3....1..1..0..1..2....2..3..1..3..2
..2..2..1..3..3....0..0..1..2..2....1..1..1..0..0....2..3..0..3..2

Row 1 is A059306

A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

Original entry on oeis.org

1, 10, 33, 88, 145, 330, 385, 736, 945, 1450, 1441, 2904, 2353, 3850, 4785, 6016, 5185, 9450, 7201, 12760, 12705, 14410, 12673, 24288, 18625, 23530, 26001, 33880, 25201, 47850, 30721, 48640, 47553, 51850, 55825, 83160, 51985, 72010, 77649, 106720

Offset: 1

David W. Wilson

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A005353, A059306, A062801, A069097, A102631, A240547.

f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 22 2020 *)

a(n)=if(n<1, 0, direuler(p=2, n, (1-p*X)/((1-p^2*X)*(1-p^3*X)))[n])

a(n)=local(c=0); forvec(x=vector(4,k,[1,n]),c+=((x[1]*x[2]-x[3]*x[4])%n==0)); c

From Vladeta Jovovic, Apr 22 2002: (Start)
a(n) = n^4 - A005353(n).
Multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1). (End)
Dirichlet g.f.: zeta(s-2)*zeta(s-3)/zeta(s-1).
A102631(n) | a(n). - R. J. Mathar, Mar 30 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / (24*Zeta(3)). - Vaclav Kotesovec, Jan 31 2019
From Piotr Rysinski, Sep 11 2020: (Start)
a(n) = n * A069097(n).
Proof: a(n) is multiplicative with a(p^e) = p^(2*e - 1)*(p^(e+1) + p^e - 1), A069097(n) is multiplicative with A069097(p^e) = p^(e-1)*(p^e*(p+1)-1), so a(p^e) = p^e*A069097(p^e). (End)

A187624 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 non-center subblock determinant nonzero and the array of 2X2 subblock determinants antisymmetric under 90 degree rotation.

Original entry on oeis.org

10, 31, 24, 64, 446, 2, 113, 2924, 60, 948, 170, 12104, 390, 324452, 36, 255, 37448, 2146, 12278892, 70548, 190972, 336, 98014, 5324, 221177664, 4232368, 6090023738, 3020, 449, 221640, 16042, 1896404966

Offset: 1

R. H. Hardin Mar 12 2011

Table starts
.....10.........31.......64.......113........170...255....336.449
.....24........446.....2924.....12104......37448.98014.221640
......2.........60......390......2146.......5324.16042
....948.....324452.12278892.221177664.1896404966
.....36......70548..4232368
.190972.6090023738
...3020

			Some k=3 solutions for 5X5
..1..0..1..0..1....2..0..1..3..2....2..0..1..2..3....1..0..2..0..1
..0..2..1..2..0....0..2..0..2..0....0..2..0..2..1....0..2..1..2..2
..1..0..1..0..1....1..0..1..0..1....1..3..2..0..1....2..0..3..0..2
..3..2..2..2..3....2..2..3..2..1....0..2..0..2..1....0..2..0..2..2
..1..0..1..0..1....2..0..1..0..2....2..3..1..0..2....1..3..2..0..1

Row 1 is A059306

A187633 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with the array of 2X2 subblock determinants antisymmetric under horizontal and vertical reflection and each non-centerline 2X2 subblock nonsingular.

Original entry on oeis.org

10, 31, 24, 64, 446, 24, 113, 2924, 1648, 954, 170, 12104, 14044, 557382, 954, 255, 37448, 106892, 31903442, 672698, 194008, 336, 98014, 306320, 686589106, 33869160, 21411605018, 194008, 449, 221640, 1848504, 7836968262, 795758912

Offset: 1

R. H. Hardin Mar 12 2011

Table starts
.....10..........31.......64.......113........170.....255....336.449
.....24.........446.....2924.....12104......37448...98014.221640
.....24........1648....14044....106892.....306320.1848504
....954......557382.31903442.686589106.7836968262
....954......672698.33869160.795758912
.194008.21411605018
.194008

			Some k=3 solutions for 5X5
..2..0..2..1..2....2..0..2..0..2....2..0..2..1..2....2..0..2..1..3
..0..1..0..1..0....0..3..2..3..0....0..2..0..2..0....0..3..0..3..3
..3..1..3..1..3....2..0..2..0..2....2..1..2..0..2....2..0..2..0..2
..0..1..0..1..0....0..3..0..3..0....0..2..0..2..2....1..3..0..3..3
..2..2..2..1..2....2..3..2..0..2....2..2..2..0..2....3..3..2..1..3

Row 1 is A059306

Row 2 is A187625

A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant.

Original entry on oeis.org

0, 6, 48, 168, 480, 966, 2016, 3360, 5616, 8550, 13200, 17832, 26208, 34566, 45840, 59520, 78336, 95526, 123120, 147240, 181776, 219846, 267168, 307488, 372000, 433446, 505440, 580776, 682080, 762150, 892800, 999936, 1138368, 1284486

Offset: 1

N. J. A. Sloane, R. K. Guy

T. Brenner, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Richard K. Guy, Letter to N. J. A. Sloane, Sep 1989.

Cf. A020478, A059306, A062801.

Table[cnt=0; Do[m={{a, b}, {c, d}}; If[Det[m, Modulus->p] > 0, cnt++ ], {a, 0, p-1}, {b, 0, p-1}, {c, 0, p-1}, {d, 0, p-1}]; cnt, {p, 37}] (* T. D. Noe, Jan 12 2006 *)
f[p_, e_] := p^(2*e - 1)*(p^(e + 1) + p^e - 1); a[1] = 0; a[n_] := n^4 - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2023 *)

a(n) = {my(f = factor(n), p, e); n^4 - prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  p^(2*e - 1)*(p^(e + 1) + p^e - 1));} \\ Amiram Eldar, Oct 31 2023

a(n) = n^4 - A020478(n).

For prime n, a(n) = (n^2-1)(n-1)n. - T. D. Noe, Jan 12 2006

More terms from T. D. Noe, Jan 12 2006

A059976 Number of 3 X 3 determinants with elements from {0,...,n} and having the value zero.

Original entry on oeis.org

1, 338, 6891, 49246, 228737, 716214, 2110081, 4663844, 10289331, 19945864, 37518971, 61582884, 109478509, 165518210, 259500567, 393804256, 586056347, 802076580, 1171487431, 1550250662, 2174383653, 2909581936, 3842464925, 4838646900, 6528085915, 8177178662, 10347435783, 12932799472

Offset: 0

Vladeta Jovovic, Mar 05 2001

nonn

Symmetry is not taken into account, i.e., symmetric solutions are counted more than once.

Cf. A059306.

More terms from John W. Layman, Nov 21 2002

More terms from Sean A. Irvine, Oct 16 2022

cp's OEIS Frontend

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

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A062801 Number of 2 X 2 non-singular integer matrices with entries from {0,...,n}.

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A134506 Number of 2 X 2 singular integer matrices with elements from {1,...,n}.

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A187521 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with the array of 2X2 subblock determinants antisymmetric and no off-diagonal 2X2 subblock determinant zero.

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A187705 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 subblock off diagonal and antidiagonal nonsingular and the array of 2X2 subblock determinants antisymmetric about the diagonal and antidiagonal.

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A020478 Number of singular 2 X 2 matrices over Z(n) (i.e., with determinant = 0).

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Mathematica

PARI

PARI

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A187624 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with each 2X2 non-center subblock determinant nonzero and the array of 2X2 subblock determinants antisymmetric under 90 degree rotation.

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A187633 T(n,k)=Number of (n+1)X(n+1) 0..k arrays with the array of 2X2 subblock determinants antisymmetric under horizontal and vertical reflection and each non-centerline 2X2 subblock nonsingular.

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A005353 Number of 2 X 2 matrices with entries mod n and nonzero determinant.

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