A280588 -id:A280588 - 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

A280934 Number of 2 X 2 matrices with all elements in {0,..,n} and (sum of terms) = permanent.

Original entry on oeis.org

1, 1, 4, 36, 52, 76, 92, 116, 136, 160, 176, 208, 224, 248, 272, 300, 316, 348, 364, 396, 420, 444, 460, 500, 520, 544, 568, 600, 616, 656, 672, 704, 728, 752, 776, 820, 836, 860, 884, 924, 940, 980, 996, 1028, 1060, 1084, 1100, 1148, 1168, 1200, 1224, 1256, 1272, 1312, 1336, 1376, 1400, 1424, 1440, 1496, 1512, 1536

Offset: 0

Indranil Ghosh, Jan 11 2017

nonn

a(n) mod 4 = 0 for n > 1.

a(n) is also the number of 2 X 2 matrices with all elements in {-1,..,n-1} and permanent = 2. - Chai Wah Wu, Jan 11 2017

			For n = 3, the possible matrices are [0,0,0,0], [0,2,2,0], [0,2,3,1],[0,3,2,1], [0,3,3,3], [1,2,3,0], [1,2,3,1], [1,2,3,2], [1,2,3,3], [1,3,2,0], [1,3,2,1], [1,3,2,2], [1,3,2,3], [2,0,0,2], [2,0,1,3], [2,1,0,3], [2,1,1,3], [2,1,2,3], [2,1,3,3], [2,2,1,3], [2,2,2,2], [2,2,3,1], [2,3,1,3], [2,3,2,1], [3,0,1,2], [3,0,3,3], [3,1,0,2], [3,1,1,2], [3,1,2,2], [3,1,3,2], [3,2,1,2], [3,2,3,1], [3,3,0,3], [3,3,1,2], [3,3,2,1] and [3,3,3,0]. There are 36 possibilities.
Here each of the matrices 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, for n = 3, a(n) = 36.

Indranil Ghosh and Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms for n = 0..200 from Indranil Ghosh)
Chai Wah Wu, Proof of formula a(n) = a(n-1) + 4*tau(n+1) + 8 for n > 3.

Cf. A000005, A210374, A280588.

def t(n):
    s=0
    for a in range(n+1):
        for b in range(n+1):
            for c in range(n+1):
                for d in range(n+1):
                    if (a+b+c+d)==(a*d+b*c):
                        s+=1
    return s
for i in range(201):
    print(str(i)+" "+str(t(i)))

from sympy import divisor_count
A280934_list = [1,1,4,36]
for i in range(4,100):
    A280934_list.append(A280934_list[-1]+4*divisor_count(i+1)+8) # Chai Wah Wu, Jan 11 2017

a(n) = a(n-1) + 4*A000005(n+1) + 8 for n > 3. - Chai Wah Wu, Jan 11 2017

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)}")

A280914 Number of 2 X 2 matrices with all terms in {-n,...,0,...,n} and (sum of terms) = permanent.

Original entry on oeis.org

1, 21, 52, 172, 268, 428, 588, 812, 1004, 1324, 1580, 1900, 2252, 2668, 2988, 3532, 3916, 4460, 5004, 5548, 6028, 6764, 7308, 8044, 8716, 9548, 10156, 11116, 11852, 12620, 13548, 14444, 15244, 16524, 17228, 18380, 19340, 20588, 21548

Offset: 0

Indranil Ghosh, Jan 18 2017

nonn

a(n) mod 4 = 0 for n > 1.

			For n = 4, a few of the possible matrices are [-4,-3,-2,3], [-4,-3,3,-1], [-4,-2,-3,3], [-4,-2,2,0], [-3,4,-1,-1], [-3,4,3,2], [-2,-4,0,2], [-2,-4,3,-3], [-1,4,1,0], [-1,4,3,3], [0,-4,0,4], [0,-4,1,-1], [0,-3,0,3], [1,2,3,0], [1,2,3,1], [1,2,3,2], [1,2,3,3], [1,2,3,4], [1,3,2,-4], [1,3,2,-3], [2,-1,0,1],... There are 268 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] and d = M[2][2]. So, a(4) = 268.

David Radcliffe, Table of n, a(n) for n = 0..10000 (terms 0..168 from Indranil Ghosh).

Cf. A280588, A280934, A281194.

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(0,169):
    print(f"{i} {t(i)}")

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

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A280934 Number of 2 X 2 matrices with all elements in {0,..,n} and (sum of terms) = permanent.

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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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A280914 Number of 2 X 2 matrices with all terms in {-n,...,0,...,n} and (sum of terms) = permanent.

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Python