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.

User: Vincent J. Matsko

Vincent J. Matsko's wiki page.

Vincent J. Matsko has authored 6 sequences.

A271316 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-cube, 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 8, 8, 1, 24, 36, 14, 1, 64, 128, 88, 24, 1, 160, 400, 400, 200, 42, 1, 384, 1152, 1520, 1120, 444, 76, 1, 896, 3136, 5152, 5040, 2968, 980, 142, 1, 2048, 8192, 16128, 19712, 15456, 7616, 2160, 272, 1, 4608, 20736, 47616, 69888, 68544, 45024, 19104, 4752, 530, 1, 10240, 51200, 134400, 230400, 271488, 223104, 126240, 47040, 10420, 1044, 1
Offset: 0

Views

Author

Vincent J. Matsko, Apr 03 2016

Keywords

Examples

			Triangle begins:
1;
2, 1;
8, 8, 1;
24, 36, 14, 1;
64, 128, 88, 24, 1;
...
Row 2 describes an octagon: 8 vertices and 8 edges.
Row 3 describes a truncated cube: 24 vertices, 36 edges, and 14 faces.

Links

Crossrefs

Cf. A038207 (n-cube).

Programs

  • Mathematica
    Flatten[Table[
  CoefficientList[
   D[1 + Exp[(x + 2) z] + ( Exp[2 z (x + 1)] - (x + 1) Exp[2 z])/x, {z,
      k}] /. z -> 0, x], {k, 0, 10}]]

Formula

G.f. for rows (n > 0): (x+2)^n + 2^n*(x+1)*((x+1)^(n-1)-1)/x.
O.g.f: 1 + 1/(1-(x+2)*y) + 1/(x*(1-2*y*(x+1))) - (x+1)/(x*(1-2*y)).
E.g.f: 1 + exp((x+2)*z) + (exp(2*z*(x+1))-(x+1)*exp(2*z))/x.

A271238 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a completely truncated n-dimensional simplex, 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 6, 12, 8, 1, 10, 30, 30, 10, 1, 15, 60, 80, 45, 12, 1, 21, 105, 175, 140, 63, 14, 1, 28, 168, 336, 350, 224, 84, 16, 1, 36, 252, 588, 756, 630, 336, 108, 18, 1, 45, 360, 960, 1470, 1512, 1050, 480, 135, 20, 1, 55, 495, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1
Offset: 0

Views

Author

Vincent J. Matsko, Apr 02 2016

Keywords

Comments

The complete truncation of a 1-dimensional segment is also a 1-dimensional segement (rather than degenerating to a point).

Examples

			Triangle begins:
1;
2, 1;
3, 3, 1;
6, 12, 8, 1;
10, 30, 30, 10, 1;
...
Row 2 describes a triangle.
Row 3 describes an octahedron.

Crossrefs

Cf. A259477 (partially-truncated simplex).

Programs

  • Mathematica
    Flatten[Table[
  CoefficientList[
   D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x +
      1 - (x + 1) z ((z + 2)*Exp[z] - 2)/2, {z, k}] /. z -> 0, x], {k, 0,
   10}]]

Formula

G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n-binomial(n+1,2)*(x+1).
O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)*y)*(1-y))+1+y*(x+1)*(1-1/(1-y)^3).
E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1 - (x+1)*z*((z+2)*exp(z)-2)/2.

A259569 Triangle T(n,k) read by rows, where T(n,k) is the number of k-dimensional faces of the polytope that is the convex hull of all permutations of the list (0,1,...,1,2), where there are n - 1 ones, for n > 0. T(0,0) is 1.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 12, 24, 14, 1, 20, 60, 70, 30, 1, 30, 120, 210, 180, 62, 1, 42, 210, 490, 630, 434, 126, 1, 56, 336, 980, 1680, 1736, 1008, 254, 1, 72, 504, 1764, 3780, 5208, 4536, 2286, 510, 1, 90, 720, 2940, 7560, 13020, 15120, 11430, 5100, 1022, 1, 110, 990, 4620, 13860, 28644, 41580, 41910, 28050, 11242, 2046, 1
Offset: 0

Views

Author

Vincent J. Matsko, Jun 30 2015

Keywords

Comments

It appears that these integers, with sign changes, are also in A138106.

Examples

			Triangle begins:
   1;
   2,  1;
   6,  6,  1;
  12, 24, 14,  1;
  20, 60, 70, 30,  1;
  ...
Row 2 describes a regular hexagon.
Row 3 describes the cuboctahedron.

Crossrefs

Row sums give A101052(n+1).
Cf. A138106.

Programs

  • Maple
    T:= (n, k)-> `if`(n=k, 1, binomial(n+1, k+2)*(4*2^k-2)):
seq(seq(T(n,k), k=0..n), n=0..10);
  • Mathematica
    Join @@ (CoefficientList[#,
     x] & /@ (Expand[
       D[((1 + 2 x) Exp[z (1 + 2 x)] - 2 (1 + x) Exp[z (1 + x)] + Exp[z] +
            x^2 Exp[z x])/x^2, {z, #}] /. z -> 0] & /@ Range[0, 10]))

Formula

T(n,n) = 1, n >= 0.
T(n,n-1) = 2^(n+1)-2, n > 0.
T(n,0) = n(n+1), n > 0.
T(n,k) = (n+1)*T(n-1,k)/(n-k-1), 0 <= k < n-1, n >= 2.
E.g.f.: ((2*x+1)*exp(z*(2*x+1)) - 2*(x+1)*exp(z*(x+1)) + x^2*exp(z*x)+exp(z))/x^2
Conjecture: Sum_{k=0..n-1} T(n,k)*x^(n-k-1) = x^(n+1) - 2(x+1)^(n+1) + (x+2)^(n+1). - Kevin J. Gomez, Jul 25 2017
T(n,n) = 1; T(n,k) = binomial(n+1,k+2)*(4*2^k - 2) for 0 <= k < n. - Aadesh Tikhe, May 25 2024

A259477 Triangle of numbers where T(n,k) is the number of k-dimensional faces on a partially truncated n-dimensional simplex, 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 12, 18, 8, 1, 20, 40, 30, 10, 1, 30, 75, 80, 45, 12, 1, 42, 126, 175, 140, 63, 14, 1, 56, 196, 336, 350, 224, 84, 16, 1, 72, 288, 588, 756, 630, 336, 108, 18, 1, 90, 405, 960, 1470, 1512, 1050, 480, 135, 20, 1, 110, 550, 1485, 2640, 3234, 2772, 1650, 660, 165, 22, 1
Offset: 0

Views

Author

Vincent J. Matsko, Jun 27 2015

Keywords

Examples

			Triangle begins:
1;
2, 1;
6, 6, 1;
12, 18, 8, 1;
20, 40, 30, 10, 1;
...

Links

Programs

  • Mathematica
    Join @@ (CoefficientList[#,
     x] & /@ (Expand[
       D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x + 1, {z, #}] /.
        z -> 0] & /@ Range[0, 10])) (* Vincent J. Matsko, Jun 30 2015 *)
Flatten[Table[
  CoefficientList[
   D[(1/(1 - (x + 1)*y)^2 - (x + 1)/(1 - y)^2)/x +
       1/((1 - (x + 1) y)*(1 - y)) + 1, {y, k}]/Factorial[k] /. y -> 0,
   x], {k, 0, 10}]] (* Vincent J. Matsko, Jul 18 2015 *)
  • PARI
    T(n,k)=max(if(k,n+1-k,n)*binomial(n+1,k+1),1)
for(n=0,10,for(k=0,n,print1(T(n,k)", "))) \\ Charles R Greathouse IV, Jun 29 2015

Formula

T(n,0) = n*(n+1), n > 0; T(n,k) = (n+1-k)*binomial(n+1,k+1), 1 <= k <= n.
E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1.
From Vincent J. Matsko, Jul 18 2015: (Start)
O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)y)*(1-y))+1.
G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n. (End)

A259053 Number of triangles with sides pairwise relatively prime and one maximal side being n; permutations of sides are not considered distinct.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 3, 2, 4, 1, 11, 2, 17, 6, 8, 9, 32, 6, 42, 12, 22, 15, 64, 12, 56, 25, 49, 25, 107, 12, 124, 51, 64, 45, 86, 27, 180, 59, 91, 45, 225, 30, 250, 81, 96, 87, 302, 55, 263, 82
Offset: 1

Views

Author

Vincent J. Matsko, Jun 29 2015

Keywords

Links

Crossrefs

First differences of A259446.

Programs

  • Mathematica
    triangles[n_] := Module[{count = 0},
   For[a = 1, a <= n, a++,
    For[b = a, b <= n, b++,
     If[(a + b > n) && CoprimeQ[a, b, n], count++]]];
   count];
triangles[#] & /@ Range[50]
  • PARI
    a(n)=if(n==1, return(1)); sum(a=3,n-2, if(gcd(a,n)==1, my(an=a*n); sum(b=max(a+1,n-a+1), n-1, gcd(an, b)==1))) \\ Charles R Greathouse IV, Jun 29 2015

A259446 Number of triangles with sides pairwise relatively prime and no greater than n; permutations of sides are not considered distinct.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 7, 11, 12, 23, 25, 42, 48, 56, 65, 97, 103, 145, 157, 179, 194, 258, 270, 326, 351, 400, 425, 532, 544, 668, 719, 783, 828, 914, 941, 1121, 1180, 1271, 1316, 1541, 1571, 1821, 1902, 1998, 2085, 2387, 2442, 2705, 2787
Offset: 1

Views

Author

Vincent J. Matsko, Jun 27 2015

Keywords

Examples

			(1,1,1) is the only triangle up to n=4; (3,4,5) is added for n=5 and n=6.

Links

Programs

  • Mathematica
    triangles[n_] := Module[{count = 0},
   For[a = 1, a <= n, a++,
    For[b = a, b <= n, b++,
     For[c = b, c <= n, c++,
      If[(a + b > c) && (b + c > a) && (c + a > b) && CoprimeQ[a, b, c],
       count++]]]];
   count];
triangles[#] & /@ Range[50]
  • PARI
    a(n)=sum(a=3,n-2, sum(b=a+1,n-1, if(gcd(a,b)==1, sum(c=b+1,min(a+b-1,n), gcd(a*b,c)==1))))+1 \\ Charles R Greathouse IV, Jun 29 2015

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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