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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A209982 Number of 2 X 2 matrices having all elements in {-n,...,n} and determinant 1.

Original entry on oeis.org

0, 20, 52, 116, 180, 308, 372, 564, 692, 884, 1012, 1332, 1460, 1844, 2036, 2292, 2548, 3060, 3252, 3828, 4084, 4468, 4788, 5492, 5748, 6388, 6772, 7348, 7732, 8628, 8884, 9844, 10356, 10996, 11508, 12276, 12660, 13812, 14388, 15156
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A171503, A196227, A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)
  • PARI
    a(n)=if(n<1, 0, 32*sum(k=1, n, eulerphi(k)) - 12) \\ Andrew Howroyd, May 05 2020

Formula

From Andrew Howroyd, May 05 2020: (Start)
a(n) = 8*A196227(n) + 4*(4*n + 1) = 8*A171503(n) - 4 for n > 0.
a(n) = -12 + 32*Sum_{k=1..n} phi(k) for n > 0. (End)

A209984 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 2.

Original entry on oeis.org

0, 4, 92, 156, 284, 412, 604, 796, 1052, 1244, 1628, 1948, 2204, 2588, 3164, 3420, 3932, 4444, 5020, 5596, 6108, 6492, 7452, 8156, 8668, 9308, 10460, 11036, 11804, 12700, 13468, 14428, 15452, 16092, 17628, 18396, 19164, 20316, 22044
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Examples

			a(1) counts these matrices (in reduced notation):
(-1,-1,1,-1), (-1,1,-1,-1), (1,-1,1,1), (1,1,-1,1)

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A209986 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 3.

Original entry on oeis.org

0, 0, 24, 176, 240, 368, 528, 720, 848, 1232, 1360, 1680, 2000, 2384, 2576, 3216, 3472, 3984, 4368, 4944, 5200, 6160, 6480, 7184, 7824, 8464, 8848, 10000, 10384, 11280, 11920, 12880, 13392, 14992, 15504, 16272, 17040, 18192, 18768, 20688
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A209988 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 4.

Original entry on oeis.org

0, 0, 52, 116, 364, 492, 684, 876, 1260, 1452, 1836, 2156, 2668, 3052, 3628, 3884, 4652, 5164, 5740, 6316, 7340, 7724, 8684, 9388, 10156, 10796, 11948, 12524, 14060, 14956, 15724, 16684, 18220, 18860, 20396, 21164, 22700, 23852, 25580
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A209990 Number of 2 X 2 matrices having all elements in {-n,...n} and determinant 5.

Original entry on oeis.org

0, 0, 8, 64, 128, 408, 472, 664, 792, 984, 1272, 1592, 1720, 2104, 2296, 2872, 3128, 3640, 3832, 4408, 4984, 5368, 5688, 6392, 6648, 7928, 8312, 8888, 9272, 10168, 10744, 11704, 12216, 12856, 13368, 15096, 15480, 16632, 17208, 17976
Offset: 0

Views

Author

Clark Kimberling, Mar 17 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    (See the Mathematica section at A209981.)

A210371 Number of 2 X 2 matrices with all elements in {0,1,...,n} and nonnegative even determinant.

Original entry on oeis.org

1, 10, 48, 112, 285, 490, 968, 1448, 2465, 3410, 5280, 6904, 10021, 12610, 17400, 21312, 28321, 33866, 43704, 51336, 64661, 74898, 92416, 105680, 128297, 145234, 173712, 194928, 230333, 256410, 299776
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 30;
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, 0, z1}] (* A210371 *)
Table[v[n], {n, 0, z1}] (* A210372 *)
Table[w[n], {n, 0, z1}] (* A210373 *)

Formula

a(n) = (A210369(n) + A059306(n))/2. - Chai Wah Wu, Nov 27 2016

A210372 Number of 2 X 2 matrices with all elements in {0,1,...,n} and positive even determinant.

Original entry on oeis.org

0, 0, 17, 48, 172, 320, 713, 1112, 2016, 2840, 4561, 6056, 8964, 11400, 15977, 19648, 26400, 31744, 41257, 48664, 61620, 71512, 88689, 101680, 123800, 140376, 168449, 189232, 224108, 249840, 292545
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 30;
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, 0, z1}] (* A210371 *)
Table[v[n], {n, 0, z1}] (* A210372 *)
Table[w[n], {n, 0, z1}] (* A210373 *)

Formula

a(n) = (A210369(n) - A059306(n))/2. - Chai Wah Wu, Nov 27 2016

Extensions

Offset corrected by Chai Wah Wu, Nov 27 2016

A210373 Number of 2 X 2 matrices with all elements in {0,1,...,n} and positive odd determinant.

Original entry on oeis.org

0, 3, 8, 48, 84, 243, 360, 768, 1040, 1875, 2400, 3888, 4788, 7203, 8624, 12288, 14400, 19683, 22680, 30000, 34100, 43923, 49368, 62208, 69264, 85683, 94640, 115248, 126420, 151875, 165600
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

See A210000 for a guide to related sequences.

Links

Crossrefs

Cf. A210000.

Programs

  • Mathematica
    a = 0; b = n; z1 = 30;
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, 0, z1}] (* A210371 *)
Table[v[n], {n, 0, z1}] (* A210372 *)
Table[w[n], {n, 0, z1}] (* A210373 *)

Formula

From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = A210370(n)/2.
a(n) = (2*n + 1 -(-1)^n)^2*(6*n + 5 -(-1)^n)*(2*n + 3 + (-1)^n)/256
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 > 8.
G.f.: -x*(3*x^5 + 17*x^4 + 16*x^3 + 28*x^2 + 5*x + 3)/((x - 1)^5*(x + 1)^4). (End)

A210374 Number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n+2.

Original entry on oeis.org

0, 4, 19, 40, 68, 104, 149, 204, 270, 348, 439, 544, 664, 800, 953, 1124, 1314, 1524, 1755, 2008, 2284, 2584, 2909, 3260, 3638, 4044, 4479, 4944, 5440, 5968, 6529, 7124, 7754, 8420, 9123, 9864, 10644, 11464, 12325, 13228, 14174, 15164
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Comments

A210374 is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3n-2.
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 + x + y + z, {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 + 2], {n, 0, z1}]    (* A210374 *)
Table[c[n, 3 n - 2], {n, 0, z1}]  (* A210374 *)

Formula

Conjectures from Colin Barker, Dec 07 2017: (Start)
G.f.: x*(4 + 3*x - 12*x^2 + 6*x^3) / (1 - x )^4.
a(n) = (-36 + 47*n + 12*n^2 + n^3) / 6 for n>0.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A210379 Number of 2 X 2 matrices with all terms in {0,1,...,n} and odd trace.

Original entry on oeis.org

0, 8, 36, 128, 300, 648, 1176, 2048, 3240, 5000, 7260, 10368, 14196, 19208, 25200, 32768, 41616, 52488, 64980, 80000, 97020, 117128, 139656, 165888, 195000, 228488, 265356, 307328, 353220, 405000, 461280, 524288, 592416, 668168
Offset: 0

Views

Author

Clark Kimberling, Mar 20 2012

Keywords

Examples

			Writing the matrices as 4-letter words, the 8 for n=1 are as follows:
1000, 1100, 1010, 1110, 0001, 0011, 0101, 0111

Links

Crossrefs

Cf. A210000, A210378.
See A210000 for a guide to related sequences.

Programs

  • Mathematica
    a = 0; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w + z, {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, 0, 2*n}]
v[n_] := Sum[c[n, 2 k - 1], {k, 1, 2*n - 1}]
Table[u[n], {n, 0, z1}] (* A210378 *)
Table[v[n], {n, 0, z1}] (* A210379 *)

Formula

a(n) + A210378(n) = (n+1)^4.
From Chai Wah Wu, Nov 27 2016: (Start)
a(n) = (n + 1)^2*((n + 1)^2 - (2*n + 1 -(-1)^n)^2/16 - (2*n + 3 + (-1)^n)^2/16).
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 7.
G.f.: -4*x*(2*x^4 + 5*x^3 + 10*x^2 + 5*x + 2)/((x - 1)^5*(x + 1)^3). (End)
From Amiram Eldar, Mar 15 2024: (Start)
a(n) = (n+1)^2*floor((n+1)^2/2).
Sum_{n>=1} 1/a(n) = Pi^4/720 + (10-Pi^2)/4. (End)
Previous Showing 11-20 of 100 results. Next

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