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.

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A211156 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and even nonnegative determinant.

Original entry on oeis.org

37, 293, 817, 2513, 4677, 10149, 15873, 28545, 40581, 65093, 86769, 128977, 164581, 231173, 285953, 385153, 464357, 605477, 715889, 909201, 1058501, 1315237, 1510721, 1844289, 2095429
Offset: 1

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.

Programs

  • 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 *)

A211157 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and positive even determinant.

Original entry on oeis.org

4, 164, 528, 1968, 3844, 8836, 14144, 26176, 37540, 61188, 82192, 123120, 157924, 223268, 276608, 374272, 452420, 591524, 700752, 891760, 1038980, 1293700, 1487744, 1818112, 2067172
Offset: 1

Views

Author

Clark Kimberling, Apr 05 2012

Keywords

Comments

For a guide to related sequences, see A210000.

Crossrefs

Cf. A210000.

Programs

  • 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 *)

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

A209979 Number of unimodular 2 X 2 matrices having all elements in {1,2,...,n}.

Original entry on oeis.org

0, 0, 4, 16, 28, 56, 68, 112, 140, 184, 212, 288, 316, 408, 452, 512, 572, 696, 740, 880, 940, 1032, 1108, 1280, 1340, 1496, 1588, 1728, 1820, 2040, 2100, 2336, 2460, 2616, 2740, 2928, 3020, 3304, 3444, 3632, 3756, 4072, 4164, 4496, 4652, 4840
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

Equivalently, the number of 2 X 2 matrices having all elements in {1,2,...,n} and having an inverse whose elements are all integers.
See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A196227, A209978, A210000.

Programs

  • Mathematica
    (See the Mathematica section at A210000.)

Formula

a(n) = 2*A196227(n) = 4*A209978(n). - Andrew Howroyd, May 05 2020

A210001 Number of 2 X 2 matrices having all elements in {1,2,...,n} and determinant 3.

Original entry on oeis.org

0, 0, 1, 7, 11, 23, 39, 59, 71, 115, 127, 163, 199, 243, 263, 339, 367, 427, 471, 539, 567, 683, 719, 803, 879, 955, 999, 1139, 1183, 1291, 1367, 1483, 1543, 1739, 1799, 1891, 1983, 2123, 2191, 2427, 2487, 2643, 2759, 2923, 2999, 3187, 3271
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000, A209978.

Programs

  • Mathematica
    (See the Mathematica section at A209978.)

A210002 Number of 2 X 2 matrices having all elements in {1,2,...,n} and determinant 4.

Original entry on oeis.org

0, 0, 0, 4, 16, 26, 44, 62, 104, 122, 164, 198, 256, 298, 364, 390, 480, 538, 604, 670, 792, 834, 948, 1030, 1120, 1194, 1332, 1398, 1584, 1690, 1780, 1894, 2080, 2154, 2340, 2430, 2616, 2754, 2964, 3054, 3240, 3394, 3532, 3694, 4008, 4098
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209978.)

A210027 Number of 2 X 2 matrices having all elements in {1,2,...,n} and determinant 5.

Original entry on oeis.org

0, 0, 0, 3, 9, 23, 27, 47, 59, 79, 111, 147, 159, 203, 223, 291, 319, 379, 399, 467, 535, 579, 615, 699, 727, 883, 927, 995, 1039, 1147, 1215, 1331, 1391, 1467, 1527, 1739, 1783, 1923, 1991, 2083, 2223, 2379, 2423, 2587, 2663, 2875, 2959, 3139
Offset: 0

Views

Author

Clark Kimberling, Mar 16 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209978.)

A210282 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant n.

Original entry on oeis.org

1, 3, 12, 21, 43, 45, 102, 75, 161, 156, 234, 163, 433, 221, 424, 460, 608, 359, 876, 447, 1049, 830, 942, 637, 1784, 932, 1304, 1306, 1905, 997, 2648, 1137, 2428, 1920, 2164, 2042, 3980, 1595, 2694, 2618, 4378, 1961, 4900, 2163, 4423, 4283
Offset: 0

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 45;
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, n], {n, 0, z1}]  (* A210282 *)

A210286 Number of 2 X 2 matrices with all elements in {0,1,...,n} and determinant = trace.

Original entry on oeis.org

1, 3, 10, 18, 28, 42, 56, 74, 96, 118, 140, 170, 200, 234, 272, 302, 336, 386, 432, 482, 536, 578, 624, 690, 752, 810, 876, 938, 1000, 1082, 1156, 1234, 1328, 1402, 1476, 1558, 1632, 1730, 1840, 1926, 2008, 2122, 2228, 2338, 2464, 2554
Offset: 0

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w + z - 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}]    (* A210286 *)

A210288 Number of 2 X 2 matrices with all elements in {0,1,...,n} and permanent = trace.

Original entry on oeis.org

1, 6, 19, 29, 41, 51, 65, 75, 89, 101, 115, 125, 143, 153, 167, 181, 197, 207, 225, 235, 253, 267, 281, 291, 313, 325, 339, 353, 371, 381, 403, 413, 431, 445, 459, 473, 497, 507, 521, 535, 557, 567, 589, 599, 617, 635
Offset: 0

Views

Author

Clark Kimberling, Mar 19 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 45;
t[n_] := t[n] = Flatten[Table[w + z - 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}]   (* A210288 *)
Previous Showing 31-40 of 100 results. Next

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