A209981 -id:A209981 - 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

A209982 Number of 2 X 2 matrices having all elements in {-n,...,n} and determinant 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A171503, A196227, A210000.

(See the Mathematica section at A209981.)

a(n)=if(n<1, 0, 32*sum(k=1, n, eulerphi(k)) - 12) \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = 8*A196227(n) + 4*(4*n + 1) = 8*A171503(n) - 4 for n > 0.
a(n) = -12 + 32*Sum_{k=1..n} phi(k) for n > 0. (End)

A209984 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 2.

Original entry on oeis.org

0, 4, 92, 156, 284, 412, 604, 796, 1052, 1244, 1628, 1948, 2204, 2588, 3164, 3420, 3932, 4444, 5020, 5596, 6108, 6492, 7452, 8156, 8668, 9308, 10460, 11036, 11804, 12700, 13468, 14428, 15452, 16092, 17628, 18396, 19164, 20316, 22044

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

			a(1) counts these matrices (in reduced notation):
(-1,-1,1,-1), (-1,1,-1,-1), (1,-1,1,1), (1,1,-1,1)

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A210000.

(See the Mathematica section at A209981.)

A209986 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 3.

Original entry on oeis.org

0, 0, 24, 176, 240, 368, 528, 720, 848, 1232, 1360, 1680, 2000, 2384, 2576, 3216, 3472, 3984, 4368, 4944, 5200, 6160, 6480, 7184, 7824, 8464, 8848, 10000, 10384, 11280, 11920, 12880, 13392, 14992, 15504, 16272, 17040, 18192, 18768, 20688

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A210000.

(See the Mathematica section at A209981.)

A209988 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 4.

Original entry on oeis.org

0, 0, 52, 116, 364, 492, 684, 876, 1260, 1452, 1836, 2156, 2668, 3052, 3628, 3884, 4652, 5164, 5740, 6316, 7340, 7724, 8684, 9388, 10156, 10796, 11948, 12524, 14060, 14956, 15724, 16684, 18220, 18860, 20396, 21164, 22700, 23852, 25580

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A210000.

(See the Mathematica section at A209981.)

A209990 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 5.

Original entry on oeis.org

0, 0, 8, 64, 128, 408, 472, 664, 792, 984, 1272, 1592, 1720, 2104, 2296, 2872, 3128, 3640, 3832, 4408, 4984, 5368, 5688, 6392, 6648, 7928, 8312, 8888, 9272, 10168, 10744, 11704, 12216, 12856, 13368, 15096, 15480, 16632, 17208, 17976

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A210000.

(See the Mathematica section at A209981.)

A211148 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive determinant.

Original entry on oeis.org

24, 248, 1056, 3008, 6904, 13624, 24448, 40576, 63640, 95288, 137632, 192384, 262392, 349688, 457088, 587520, 744344, 930104, 1149152, 1404160, 1699640, 2039544, 2428352, 2869312, 3368472, 3929912, 4558688, 5259712, 6039480

Offset: 1

Clark Kimberling, Apr 04 2012

nonn

For a guide to related sequences, see A210000.

Cf. A210000.

a = -n; b = n; z1 = 30;
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, 1, z1}]   (* A209981 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 1, m}]
t = Table[c1[n, 2*n^2], {n, 1, z1}]   (* A211148 *)
2 t   (* A211149 *)
t/8   (* integers *)

A211149 Number of 2 X 2 nonsingular matrices having all terms in {-n,...,0,...,n}.

Original entry on oeis.org

48, 496, 2112, 6016, 13808, 27248, 48896, 81152, 127280, 190576, 275264, 384768, 524784, 699376, 914176, 1175040, 1488688, 1860208, 2298304, 2808320, 3399280, 4079088, 4856704, 5738624, 6736944, 7859824, 9117376, 10519424

Offset: 1

Clark Kimberling, Apr 04 2012

nonn

A211149(n) + A209981(n) = (2n+1)^4 for n>0.
It appears that 16 divides A211149(n).
For a guide to related sequences, see A210000.

Cf. A210000.

a = -n; b = n; z1 = 30;
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, 1, z1}]   (* A209981 *)
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 1, m}]
t = Table[c1[n, 2*n^2], {n, 1, z1}]   (* A211148 *)
2 t   (* A211149 *)
t/8   (* integers *)

A281194 Number of 2 X 2 matrices with all terms in {-n,..,0,..,n} and (sum of terms) = determinant.

Original entry on oeis.org

1, 31, 111, 271, 479, 831, 1167, 1711, 2239, 2975, 3631, 4687, 5407, 6655, 7759, 9135, 10367, 12127, 13231, 15375, 16991, 19135, 20879, 23471, 25215, 27999, 30319, 33167, 35359, 39167, 41039, 44975, 47615, 50975, 54511, 58767, 61791, 66239, 69391

Offset: 0

Indranil Ghosh, Jan 17 2017

nonn

			For n = 3, few of the possible matrices are [-3,-3,-3,0], [-3,-3,-1,1], [-3,-3,1,2], [-3,-3,3,3], [-3,-2,-1,1], [-3,-2,3,2], [-3,-1,-3,1], [-3,-1,-2,1], [-3,-1,-1,1], [-3,-1,0,1], [-3,-1,1,1], [-3,-1,2,1], [-3,-1,3,1], [-3,0,-1,1], [2,0,0,2], [2,0,1,3], [2,1,-3,-3], [2,1,-2,-1], [2,1,-1,1], [3,3,0,3],...There are 271 possibilities.
Here each of the matrices M is defined as M = [a,b;c,d] where a = M[1][1], b = M[1][2], c = M[2][1], d = M[2][2]. So, a(3) = 271.

David Radcliffe, Table of n, a(n) for n = 0..10000 (terms 0..186 from Indranil Ghosh).

Cf. A280588, A280934, A209981.

a(n)=sum(a=-n,n, sum(d=-n,n, my(t=a*d+a+d); sum(b=-n,n, if(b==-1, if(t==-1, 2*n+1, 0), my(c=(t-b)/(b+1)); denominator(c)==1 && c<=n && c>=-n)))) \\ Charles R Greathouse IV, Jan 17 2017

def t(n):
    s=0
    for a in range(-n, n+1):
        for b in range(-n, n+1):
            for c in range(-n, n+1):
                for d in range(-n, n+1):
                    if (a+b+c+d)==(a*d-b*c):
                        s+=1
    return s
for i in range(0, 187):
    print(f"{i} {t(i)}")

A209983 (A209982)/2.

Original entry on oeis.org

0, 10, 26, 58, 90, 154, 186, 282, 346, 442, 506, 666, 730, 922, 1018, 1146, 1274, 1530, 1626, 1914, 2042, 2234, 2394, 2746, 2874, 3194, 3386, 3674, 3866, 4314, 4442, 4922, 5178, 5498, 5754, 6138, 6330, 6906, 7194, 7578, 7834

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

Cf. A210000.

(See the Mathematica section at A209981.)

cp's OEIS Frontend

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

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A209982 Number of 2 X 2 matrices having all elements in {-n,...,n} and determinant 1.

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A209984 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 2.

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A209986 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 3.

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A209988 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 4.

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A209990 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 5.

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A211148 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive determinant.

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A211149 Number of 2 X 2 nonsingular matrices having all terms in {-n,...,0,...,n}.

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A281194 Number of 2 X 2 matrices with all terms in {-n,..,0,..,n} and (sum of terms) = determinant.

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