A134506 -id:A134506 - 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

A059306 Number of 2 X 2 singular integer matrices with elements from {0,...,n}.

Original entry on oeis.org

1, 10, 31, 64, 113, 170, 255, 336, 449, 570, 719, 848, 1057, 1210, 1423, 1664, 1921, 2122, 2447, 2672, 3041, 3386, 3727, 4000, 4497, 4858, 5263, 5696, 6225, 6570, 7231, 7600, 8177, 8730, 9263, 9872, 10689, 11130, 11727, 12384, 13265, 13754, 14703

Offset: 0

John W. Layman, Jan 25 2001

Vincenzo Librandi and Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms for n = 0..100 from Vincenzo Librandi)

Cf. A062801, A134506.

a[0] = 1; a[n_] := Table[{w, x, y, z} /. {ToRules[ Reduce[0 <= x <= n && 0 <= y <= n && 0 <= z <= n && w*z - x*y == 0, {x, y, z}, Integers]] }, {w, 0, n}] // Flatten[#, 1]& // Length; Table[Print[an = a[n]]; an, {n, 0, 42}] (* Jean-François Alcover, Oct 11 2013 *)

from math import gcd
def A059306(n): return (2*n+1)*(n+1) + 4*sum(gcd(i, j) for i in range(1, n+1) for j in range(i, n+1)) # David Radcliffe, Aug 13 2025

a(n) = A134506(n) + (2n+1)^2. Shi's result (see formula section in A134506) shows that a(n) = kn^2 log n + cn^2 + O(n^e) where k = 12/Pi^2, e > 547/416 = 1.3149..., and c = 4.5113... - Chai Wah Wu, Nov 28 2016

a(n) = 4*A272718(n) + 2*n^2 + 3*n + 1. - David Radcliffe, Aug 13 2025

More terms from Larry Reeves (larryr(AT)acm.org), Jan 09 2003

A209981 Number of singular 2 X 2 matrices having all elements in {-n,...,n}.

Original entry on oeis.org

1, 33, 129, 289, 545, 833, 1313, 1729, 2369, 3041, 3905, 4577, 5857, 6657, 7905, 9345, 10881, 11937, 13953, 15137, 17441, 19521, 21537, 22977, 26177, 28257, 30657, 33249, 36577, 38401, 42721, 44673, 48257, 51617, 54785, 58529, 63905

Offset: 0

Clark Kimberling, Mar 17 2012

nonn

See A210000 for a guide to related sequences.

			Among the 33 matrices counted by a(1) are these (in compact notation):
(-1,-1,-1,-1), (0,0,0,0), (1,-1,-1,1), (1,1,1,1).

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

Cf. A210000.

a = -n; b = n; z1 = 40;
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}]  (* A209981 *)
Table[c[n, 1], {n, 0, z1}]  (* A209982 *)
%/4                         (* A206258 *)
2 %                         (* A209983 *)
Table[c[n, 2], {n, 0, z1}]  (* A209984 *)
%/4                         (* A209985 *)
Table[c[n, 3], {n, 0, z1}]  (* A209986 *)
%/8                         (* A209987 *)
Table[c[n, 4], {n, 0, z1}]  (* A209988 *)
%/4                         (* A209989 *)
Table[c[n, 5], {n, 0, z1}]  (* A209990 *)
%/8                         (* A209997 *)

a(n) = 8*A134506(n) + (4*n + 1)^2. - Andrew Howroyd, May 04 2020

A209978 a(n) = A196227(n)/2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Mar 16 2012

nonn

See A210000 for a guide to related sequences.

Cf. A196227, A210000.

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)-1 + 2*numtheory[phi](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 05 2020

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 *)
%/2                         (* A209978 *)
Table[2 c[n, 1], {n, 0, z1}](* A209979 *)
Table[c[n, 2], {n, 0, z1}]  (* A197168 *)
%/2                         (* A209980 *)
Table[c[n, 3], {n, 0, z1}]  (* A210001 *)
Table[c[n, 4], {n, 0, z1}]  (* A210002 *)
Table[c[n, 5], {n, 0, z1}]  (* A210027 *)

A211066 Number of 2 X 2 matrices having all terms in {1,...,n} and nonnegative even determinant.

Original entry on oeis.org

1, 8, 28, 96, 193, 448, 728, 1360, 1985, 3264, 4420, 6696, 8641, 12296, 15360, 20896, 25361, 33344, 39636, 50680, 59289, 74056, 85376, 104728, 119377, 144032, 162588, 193568, 216585, 254880, 283096, 329656, 363881, 419856, 460804

Offset: 1

Clark Kimberling, Mar 31 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A210000.

a = 1; b = n; z1 = 35;
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]
u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
Table[u[n], {n, 1, z1}]  (* A211066 *)
Table[v[n], {n, 1, z1}]  (* A211067 *)
Table[w[n], {n, 1, z1}]  (* A211068 *)

a(n) = (A211064(n) + A134506(n))/2. - Chai Wah Wu, Nov 28 2016

A211067 Number of 2 X 2 matrices having all terms in {1,...,n} and positive even determinant.

Original entry on oeis.org

0, 2, 13, 64, 144, 362, 617, 1200, 1776, 2986, 4101, 6264, 8160, 11714, 14657, 20064, 24464, 32266, 38485, 49320, 57752, 72354, 83585, 102632, 117120, 141578, 159917, 190592, 213496, 251370, 279465, 325704, 359640, 415354, 455973

Offset: 1

Clark Kimberling, Mar 31 2012

nonn

For a guide to related sequences, see A210000.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A210000.

a = 1; b = n; z1 = 35;
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]
u[n_] := u[n] = Sum[c[n, 2 k], {k, 0, n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, n^2}]
Table[u[n], {n, 1, z1}]  (* A211066 *)
Table[v[n], {n, 1, z1}]  (* A211067 *)
Table[w[n], {n, 1, z1}]  (* A211068 *)

a(n) = (A211064(n) - A134506(n))/2. - Chai Wah Wu, Nov 28 2016

A375512 a(n) is the number of distinct compositions of four positive integers x, u, v, y (x < u <= v < y) for which x + u + v + y = n and uv = xy.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 4, 3, 0, 4, 3, 0, 6, 0, 3, 5, 0, 6, 9, 0, 0, 6, 8, 0, 9, 0, 5, 13, 0, 0, 13, 6, 7, 8, 6, 0, 11, 10, 12, 9, 0, 0, 23, 0, 0, 19, 10, 12, 15, 0, 8, 11, 18, 0, 27, 0, 0, 23, 9, 15, 18, 0, 25, 19

Offset: 0

Felix Huber, Aug 19 2024

nonn

(bin(4,0) + bin(4,2) + bin(4,4))*a(n) = 8*a(n) is the number of distinct compositions of four integers x, u, v, y (abs(x) < abs(u) <= abs(v) < abs(y)) for which abs(x) + abs(u) + abs(v) + abs(y) = n and u*v = x*y.
a(n) is also the number of 2X2 matrices having the determinant 0 whose elements [x,u;v,y] are positive integers with x < u <= v < y and x + u + v + y = n.
a(n) is also the number of distinct linear 2X2 equation systems that do not have exactly one solution and whose coefficients [x,u;v,y] are positive integers with x < u <= v < y and x + u + v + y = n.

			a(9) = 1 because only (1, 2, 2, 4) satisfies the conditions: 1 + 2 + 2 + 4 = 9 and 2*2 = 1*4.
a(24) = 4 because (1, 2, 7, 14), (1, 3, 5, 15), (2, 4, 6, 12), (3, 5, 6, 10) satisfy the conditions: 1 + 2 + 7 + 14 = 24 and 2*7 = 1*14, 1 + 3 + 5 + 15 = 24 and 3*5 = 1*15, 2 + 4 + 6 + 12 = 24 and 4*6 = 2*12, 3 + 5 + 6 + 10 = 24 and 5*6 = 3*10.
See also linked Maple code.

Felix Huber, Table of n, a(n) for n = 0..1000
Felix Huber, Maple codes

Cf. A038548, A134506, A357259.

Maple
```
See Huber link.
```

def A375512(n): return sum(1 for x in range(1,(n>>2)+1) for y in range(x+1,(n-x)//3+1) for z in range(y,(n-y>>1)+1) if xChai Wah Wu, Aug 23 2024

Conjecture: a(p) = 0 for primes p.
From Robert Israel, Aug 23 2024: (Start)
The conjecture is true, in fact for any x,y,u,v as in the definition, n has proper divisor gcd(x,u) + gcd(v,y).
Proof: Suppose x,y,u,v are positive integers with x + y + u + v = n and x*y = u*v = m.  Let g = gcd(x,u).  Then x = g*X and u = g*U where X and U are coprime.  Since X*y = U*v = m/g, we must have y = h*U and v = h*X where h = gcd(v,y).  Then n = g*X + h*U + g*U + h*X = (g+h)*(U+X).
(End)

A211056 Number of 2 X 2 nonsingular matrices having all terms in {1,...,n}.

Original entry on oeis.org

0, 10, 66, 224, 576, 1210, 2290, 3936, 6352, 9722, 14322, 20304, 28080, 37834, 49922, 64704, 82624, 103898, 129170, 158640, 192944, 232554, 278050, 329680, 388368, 454522, 528770, 611680, 704192, 806490, 919890, 1044624, 1181680

Offset: 1

Clark Kimberling, Mar 31 2012

nonn

A211056(n) + A134506(n) = n^4.

For a guide to related sequences, see A210000.

Cf. A134506, A210000, A211059.

a = 1; b = n; z1 = 35;
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]
c1[n_, m_] := c1[n, m] = Sum[c[n, k], {k, 0, m}]
Table[c1[n, n^2] - c[n, 0], {n, 1, z1}]   (* A211059 *)
2*%     (* A211056 *)

A134978 Number of 2 X 2 singular integer matrices with entries from {2,...,n}.

Original entry on oeis.org

0, 1, 6, 15, 28, 53, 74, 111, 152, 209, 246, 339, 384, 473, 582, 695, 756, 917, 986, 1175, 1340, 1493, 1578, 1855, 2008, 2193, 2398, 2683, 2792, 3185, 3302, 3603, 3880, 4129, 4446, 4943, 5084, 5365, 5698, 6231, 6388, 6973, 7138, 7615, 8172, 8517, 8698, 9431

Offset: 1

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Cf. A134506.

a[n_] := Sum[Sum[Sum[Boole[d <= n && d > 1 && b*c/d <= n && b*c > d], {d, Divisors[b*c]}], {c, 2, n}], {b, 2, n}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 25 2024, after Charles R Greathouse IV *)

a(n)=sum(b=2,n,sum(c=2,n,sumdiv(b*c,a, a<=n && a>1 && b*c/a<=n && b*c>a))) \\ Charles R Greathouse IV, Jun 17 2013

a(n) << n^(2+e) for all e > 0. - Charles R Greathouse IV, Jun 17 2013

A134505 Replace 0's with 1's in triangle A049310, and take absolute values.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 6, 1, 5, 1, 1, 1, 4, 1, 10, 1, 6, 1, 1, 1, 1, 10, 1, 15, 1, 7, 1, 1, 1, 5, 1, 20, 1, 21, 1, 8, 1, 1, 1, 1, 15, 1, 35, 1, 28, 1, 9, 1, 1, 1, 6, 1, 35, 1, 56, 1, 36, 1, 10, 1, 1

Offset: 1

Gary W. Adamson, Oct 28 2007

Row sums = A134506: (1, 2, 3, 5, 7, 11, 16, ...).

			First few rows of the triangle are:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  1,  1;
  1,  1,  3,  1,  1;
  1,  3,  1,  4,  1,  1;
  1,  1,  6,  1,  5,  1,  1;
  1,  4,  1, 10,  1,  6,  1,  1;
  1,  1, 10,  1, 15,  1,  7,  1,  1;
  ...

Cf. A049310, A134506.

Definition corrected by Eric Rowland, Jun 23 2017

a(43) = 7 corrected and more terms from Georg Fischer, Jun 07 2023

cp's OEIS Frontend

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

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A059306 Number of 2 X 2 singular integer matrices with elements from {0,...,n}.

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A209981 Number of singular 2 X 2 matrices having all elements in {-n,...,n}.

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A209978 a(n) = A196227(n)/2.

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A211066 Number of 2 X 2 matrices having all terms in {1,...,n} and nonnegative even determinant.

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A211067 Number of 2 X 2 matrices having all terms in {1,...,n} and positive even determinant.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A375512 a(n) is the number of distinct compositions of four positive integers x, u, v, y (x < u <= v < y) for which x + u + v + y = n and u*v = x*y.

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Maple

Python

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A211056 Number of 2 X 2 nonsingular matrices having all terms in {1,...,n}.

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A375512 a(n) is the number of distinct compositions of four positive integers x, u, v, y (x < u <= v < y) for which x + u + v + y = n and uv = xy.