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 1-4 of 4 results.

A131520 Number of partitions of the graph G_n (defined below) into "strokes".

Original entry on oeis.org

2, 6, 12, 22, 40, 74, 140, 270, 528, 1042, 2068, 4118, 8216, 16410, 32796, 65566, 131104, 262178, 524324, 1048614, 2097192, 4194346, 8388652, 16777262, 33554480, 67108914, 134217780, 268435510, 536870968, 1073741882, 2147483708
Offset: 1

Views

Author

Yasutoshi Kohmoto, Aug 15 2007

Keywords

Comments

G_n = {V_n, E_n}, V_n = {v_1, v_2, ..., v_n}, E_n = {v_1 v_2, v_2 v_3, ..., v_{n-1} v_n, v_n v_1}
See the definition of "stroke" in A089243.
A partition of a graph G into strokes S_i must satisfy the following conditions, where H is a digraph on G:
- Union_{i} S_i = H,
- i != j => S_i and S_j do not have a common edge,
- i != j => S_i U S_j is not a directed path,
- For all i, S_i is a dipath.
a(n) is also the number of maximal subsemigroups of the monoid of partial order preserving mappings on a set with n elements. - James Mitchell and Wilf A. Wilson, Jul 21 2017

Examples

			Figure for G_4: o-o-o-o-o Two vertices on both sides are the same.

Links

Crossrefs

Cf. A131518, A173740, A354228.

Programs

  • Magma
    [2^n + 2*(n-1): n in [1..30]]; // G. C. Greubel, Feb 13 2021
  • Mathematica
    Table[2^n + 2*(n-1), {n, 30}] (* G. C. Greubel, Feb 13 2021 *)
  • Sage
    [2^n + 2*(n-1) for n in (1..30)] # G. C. Greubel, Feb 13 2021

Formula

a(n) = 2*(n-1) + 2^n = 2*A006127(n-1).
G.f.: 2*x*(1 - x - x^2)/((1-x)^2 * (1-2*x)). - R. J. Mathar, Nov 14 2007
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3). - Wesley Ivan Hurt, May 20 2021

Extensions

More terms from Max Alekseyev, Sep 29 2007

A132735 Triangle T(n,k) = binomial(n,k) + 1 with T(n,0) = T(n,n) = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 7, 5, 1, 1, 6, 11, 11, 6, 1, 1, 7, 16, 21, 16, 7, 1, 1, 8, 22, 36, 36, 22, 8, 1, 1, 9, 29, 57, 71, 57, 29, 9, 1, 1, 10, 37, 85, 127, 127, 85, 37, 10, 1, 1, 11, 46, 121, 211, 253, 211, 121, 46, 11, 1, 1, 12, 56, 166, 331, 463, 463, 331, 166, 56, 12, 1
Offset: 0

Views

Author

Gary W. Adamson, Aug 26 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  1, 1;
  1, 3,  1;
  1, 4,  4,  1;
  1, 5,  7,  5,  1;
  1, 6, 11, 11,  6, 1;
  1, 7, 16, 21, 16, 7, 1;
  ...

Links

Crossrefs

Cf. A103451, A132736.
Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), this sequence (q=1), A173740 (q=2), A173741 (q=4), A173742 (q=6).

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 1 >;
[T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel, Feb 14 2021
  • Mathematica
    T[n_, k_]:= If[k==0||k==n, 1, Binomial[n,k] +1];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 14 2021 *)
  • Sage
    def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 1
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 14 2021

Formula

T(n, k) = A007318(n,k) + 1 - A103451(n,k), an infinite lower triangular matrix.
T(n,0) = T(n,n) = 1; T(n,k) = C(n,k) + 1 otherwise. - Franklin T. Adams-Watters, Jul 06 2009
Sum_{k=0..n} T(n, k) = 2^n + n - 1 + [n=0] = A132736(n). - G. C. Greubel, Feb 14 2021

Extensions

Corrected and extended by Franklin T. Adams-Watters, Jul 06 2009

A173741 T(n,k) = binomial(n,k) + 4 for 1 <= k <= n - 1, n >= 2, and T(n,0) = T(n,n) = 1 for n >= 0, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 7, 7, 1, 1, 8, 10, 8, 1, 1, 9, 14, 14, 9, 1, 1, 10, 19, 24, 19, 10, 1, 1, 11, 25, 39, 39, 25, 11, 1, 1, 12, 32, 60, 74, 60, 32, 12, 1, 1, 13, 40, 88, 130, 130, 88, 40, 13, 1, 1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1, 1, 15, 59, 169, 334, 466, 466, 334, 169, 59, 15, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 23 2010

Keywords

Comments

For n >= 1, row n sums to 2*A100314(n).

Examples

			Triangle begins:
  1;
  1,  1;
  1,  6,  1;
  1,  7,  7,   1;
  1,  8, 10,   8,   1;
  1,  9, 14,  14,   9,   1;
  1, 10, 19,  24,  19,  10,   1;
  1, 11, 25,  39,  39,  25,  11,   1;
  1, 12, 32,  60,  74,  60,  32,  12,  1;
  1, 13, 40,  88, 130, 130,  88,  40, 13,  1;
  1, 14, 49, 124, 214, 256, 214, 124, 49, 14, 1;
  ...

Links

Crossrefs

Cf. A100314, A103451, A156050.
Sequences of the form binomial(n, k) + q: A132823 (q=-2), A132044 (q=-1), A007318 (q=0), A132735 (q=1), A173740 (q=2), this sequence (q=4), A173742 (q=6).

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) + 4 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
  • Mathematica
    T[n_, m_] = Binomial[n, m] + 4*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
  • Maxima
    T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 4$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */
  • Sage
    def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 4
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

Formula

From Franck Maminirina Ramaharo, Dec 09 2018:(Start)
T(n,k) = A007318(n,k) + 2*(1 - A103451(n,k)).
T(n,k) = 5*A007318(n,k) - 4*A132044(n,k).
n-th row polynomial is 2*(1 - (-1)^(2^n)) + (1 + x)^n + 4*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 5*x*y^2 - 4*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (4 - 4*x + 4*x*exp(y) - 4*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 4*(n - 1 + [n=0]) = 2*A100314(n). - G. C. Greubel, Feb 13 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018

A173742 Triangle T(n,k) = binomial(n,k) + 6 with T(n,0) = T(n,n) = 1 for n >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 9, 9, 1, 1, 10, 12, 10, 1, 1, 11, 16, 16, 11, 1, 1, 12, 21, 26, 21, 12, 1, 1, 13, 27, 41, 41, 27, 13, 1, 1, 14, 34, 62, 76, 62, 34, 14, 1, 1, 15, 42, 90, 132, 132, 90, 42, 15, 1, 1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1, 1, 17, 61, 171, 336, 468, 468, 336, 171, 61, 17, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 23 2010

Keywords

Comments

For n >= 1, row n sums to A131520(n) + A008586(n).

Examples

			Triangle begins:
  1;
  1,  1;
  1,  8,  1;
  1,  9,  9,   1;
  1, 10, 12,  10,   1;
  1, 11, 16,  16,  11,   1;
  1, 12, 21,  26,  21,  12,   1;
  1, 13, 27,  41,  41,  27,  13,   1;
  1, 14, 34,  62,  76,  62,  34,  14,  1;
  1, 15, 42,  90, 132, 132,  90,  42, 15,  1;
  1, 16, 51, 126, 216, 258, 216, 126, 51, 16, 1;
  ...

Links

Crossrefs

Cf. A007318, A103451, A132044, A156050, A173740, A173741.

Programs

  • Magma
    T:= func< n,k | k eq 0 or k eq n select 1 else Binomial(n,k) +6 >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 13 2021
  • Mathematica
    T[n_, m_] = Binomial[n, m] + 6*If[m*(n - m) > 0, 1, 0];
Flatten[Table[T[n, m], {n, 0, 10}, {m, 0, n}]]
  • Maxima
    T(n,k) := if k = 0 or k = n then 1 else binomial(n, k) + 6$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 09 2018 */
  • Sage
    def T(n, k): return 1 if (k==0 or k==n) else binomial(n, k) + 6
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 13 2021

Formula

From Franck Maminirina Ramaharo, Dec 09 2018: (Start)
T(n,k) = A007318(n,k) + 6*(1 - A103451(n,k)).
T(n,k) = 7*A007318(n,k) - 6*A132044(n,k).
n-th row polynomial is 3*(1 - (-1)^(2^n)) + (1 + x)^n + 6*(x - x^n)/(1 - x).
G.f.: (1 - (1 + x)*y + 7*x*y^2 - 6*(x + x^2)*y^3)/((1 - y)*(1 - x*y)*(1 - y - x*y)).
E.g.f.: (6 - 6*x + 6*x*exp(y) - 6*exp(x*y) + (1 - x)*exp((1 + x)*y))/(1 - x). (End)
Sum_{k=0..n} T(n, k) = 2^n + 6*n - 6 + 6*[n=0]. - G. C. Greubel, Feb 13 2021

Extensions

Edited and name clarified by Franck Maminirina Ramaharo, Dec 09 2018
Showing 1-4 of 4 results.

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