cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

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

Examples

			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)

Crossrefs

Cf. A171503.
See also the very useful list of cross-references in the Comments section.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

A211158 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive odd determinant.

Original entry on oeis.org

20, 84, 528, 1040, 3060, 4788, 10304, 14400, 26100, 34100, 55440, 69264, 104468, 126420, 180480, 213248, 291924, 338580, 448400, 512400, 660660, 745844, 940608, 1051200, 1301300, 1441908, 1756944, 1932560, 2322900, 2538900, 3015680, 3277824, 3852948, 4167380
Offset: 1

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000.

Programs

  • Magma
    [n*(n+1)*(3*n+1+3*n^2-(-1)^n*(2*n+1)): n in [1..35]]; // Vincenzo Librandi, Dec 14 2016
  • Mathematica
    a = -n; b = n; z1 = 25;
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, 2*n^2}]
v[n_] := v[n] = Sum[c[n, 2 k], {k, 1, 2*n^2}]
w[n_] := w[n] = Sum[c[n, 2 k - 1], {k, 1, 2*n^2}]
u1 = Table[u[n], {n, 1, z1}] (* A211156 *)
v1 = Table[v[n], {n, 1, z1}] (* A211157 *)
w1 = Table[w[n], {n, 1, z1}] (* A211158 *)
(u1 - 1)/4 (* integers *)
v1/4 (* integers *)
w1/4 (* integers *)
Table[n*(n+1)*(3*n+1+3*n^2-(-1)^n*(2*n+1)),{n,35}] (* Vincenzo Librandi, Dec 14 2016 *)
CoefficientList[ Series[-(( 4(5 + 16x + 91x^2 + 64x^3 + 91x^4 + 16x^5 + 5x^6))/((x -1)^5 (x +1)^4)), {x, 0, 35}], x] (* or *)
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {20, 84, 528, 1040, 3060, 4788, 10304, 14400, 26100}, 36] (* Robert G. Wilson v, Dec 14 2016 *)
  • Python
    def A211158(n):
    return n*(n+1)*(3*n+1+3*n**2-(-1)**n*(2*n+1)) # Chai Wah Wu, Dec 13 2016

Formula

From Chai Wah Wu, Dec 13 2016: (Start)
For n >= 0:
a(n) = A211155(n)/2.
a(n) = n*(n + 1)*(3*n + 1 + 3*n^2 - (-1)^n*(2*n + 1)). Therefore:
a(n) = n^2*(n + 1)*(3*n + 1) if n is even,
a(n) = n*(n + 1)^2*(3*n + 2) if n is odd.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(-20*x^6 - 64*x^5 - 364*x^4 - 256*x^3 - 364*x^2 - 64*x - 20)/((x - 1)^5*(x + 1)^4). (End)
a(n) = a(-n-1). - Bruno Berselli, Dec 14 2016

A211154 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and even determinant.

Original entry on oeis.org

1, 41, 457, 1345, 4481, 8521, 18985, 30017, 54721, 78121, 126281, 168961, 252097, 322505, 454441, 562561, 759425, 916777, 1197001, 1416641, 1800961, 2097481, 2608937, 2998465, 3662401, 4162601, 5006665, 5636737, 6690881, 7471561, 8768041, 9721601, 11294977, 12445225, 14332361, 15704641
Offset: 0

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Links

Crossrefs

Cf. A210000, A211155.

Programs

  • Maple
    seq((2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n), n=1..20); # Mark van Hoeij, May 13 2013
  • Mathematica
    a = -n; b = n; z1 = 20;
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_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 1, z1}] (* A211154 *)
Table[v[n], {n, 1, z1}] (* A211155 *)
  • PARI
    a(n)=(2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n); \\ Joerg Arndt, May 14 2013

Formula

a(n) + A211155(n) = (2n+1)^4.
From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(-x^8 - 36*x^6 - 416*x^5 - 734*x^4 - 1472*x^3 - 724*x^2 - 416*x - 41)/((x - 1)^5*(x + 1)^4). (End)

Extensions

More terms from Joerg Arndt, May 14 2013
a(0)=1 prepended by Andrew Howroyd, May 05 2020
Showing 1-3 of 3 results.

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