A211059 -id:A211059 - 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

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 *)

A373723 Number of strictly totally positive 3 X 3 matrices having all terms in {1,...,n}.

Original entry on oeis.org

0, 0, 22, 597, 7178, 43090, 207494, 748801, 2321973, 6267631, 15596170, 34784307, 74017706, 147072570, 277965322, 503711791, 884612799, 1491687919, 2458600175, 3925566799, 6133712065, 9388594434, 14121653942, 20783339478, 30178942357, 43156537147, 60868287839, 84699183224, 116688767652

Offset: 1

José María Grau Ribas, Jun 15 2024

nonn

A matrix is strictly totally positive if all its minors are greater than zero.

Robin Visser, Table of n, a(n) for n = 1..40

Cf. A211059, A373724.

ispositive1[M_]:=ispositive1[M]=Union@Table[Select[Union@Flatten@Minors[M,r],(#<= 0)&]=={},{r,1,Length[M]}]=={True}; W[n_]:=W[n]=Flatten[Table[{{a11,a12,a13},{a21,a22,a23},{a31,a32,a33}},{a11,1,n},{a12,1,n},{a13,1,n},{a21,1,n},{a22,1,n},{a23,1,n},{a31,1,n},{a32,1,n},{a33,1,n}],8];  Table[Length@Select[W[n],ispositive1[#]&],{n,1,7}]

import itertools
def a(n):
    ans, W = 0, itertools.product(range(1,n+1), repeat=9)
    for w in W:
        M = Matrix(ZZ, 3, 3, w)
        if (min(M.minors(2)) > 0) and (M.det() > 0): ans += 1
    return ans  # Robin Visser, Apr 18 2025

More terms from Robin Visser, Apr 18 2025

A373724 Number of totally positive 3 X 3 matrices having all terms in {1,...,n}.

Original entry on oeis.org

1, 61, 797, 6490, 32744, 146441, 492277, 1521123, 4105795, 10194558, 22922408, 49594408, 98935110, 190221734, 350417949, 621178227, 1058404994, 1764873413, 2845696865, 4506618651, 6966717779, 10552756376, 15670141644, 22984055065, 33094853060, 47016605050, 65934960254, 91414399149

Offset: 1

José María Grau Ribas, Jun 15 2024

nonn

A matrix is totally positive if all its minors are nonnegative.

Robin Visser, Table of n, a(n) for n = 1..40

Cf. A211059, A373723.

ispositive2[M_]:=ispositive1[M]=Union@Table[Select[Union@Flatten@Minors[M,r],(#<= 0)&]=={},{r,1,Length[M]}]=={True};
W[n_]:=W[n]=Flatten[Table[{{a11,a12,a13},{a21,a22,a23},{a31,a32,a33}},{a11,1,n},{a12,1,n},{a13,1,n},{a21,1,n},{a22,1,n},{a23,1,n},{a31,1,n},{a32,1,n},{a33,1,n}],8];
Table[Length@Select[W[n],ispositive2[#]&],{n,1,6}]

import itertools
def a(n):
    ans, W = 0, itertools.product(range(1,n+1), repeat=9)
    for w in W:
        M = Matrix(ZZ, 3, 3, w)
        if (min(M.minors(2)) >= 0) and (M.det() >= 0): ans += 1
    return ans  # Robin Visser, Apr 18 2025

More terms from Robin Visser, Apr 18 2025

cp's OEIS Frontend

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

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

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A373723 Number of strictly totally positive 3 X 3 matrices having all terms in {1,...,n}.

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Mathematica

Sage

Extensions

A373724 Number of totally positive 3 X 3 matrices having all terms in {1,...,n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Sage

Extensions