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.

Previous Showing 11-13 of 13 results.

A326253 Number of sequences of distinct ordered pairs of positive integers up to n.

Original entry on oeis.org

1, 2, 65, 986410, 56874039553217, 42163840398198058854693626, 1011182700521015817607065606491025592595137, 1653481537585545171449931620186035466059689728986775126016505970
Offset: 0

Views

Author

Gus Wiseman, Jun 21 2019

Keywords

Examples

			The a(2) = 65 sequences:
  ()  (11)  (11,12)  (11,12,21)  (11,12,21,22)
      (12)  (11,21)  (11,12,22)  (11,12,22,21)
      (21)  (11,22)  (11,21,12)  (11,21,12,22)
      (22)  (12,11)  (11,21,22)  (11,21,22,12)
            (12,21)  (11,22,12)  (11,22,12,21)
            (12,22)  (11,22,21)  (11,22,21,12)
            (21,11)  (12,11,21)  (12,11,21,22)
            (21,12)  (12,11,22)  (12,11,22,21)
            (21,22)  (12,21,11)  (12,21,11,22)
            (22,11)  (12,21,22)  (12,21,22,11)
            (22,12)  (12,22,11)  (12,22,11,21)
            (22,21)  (12,22,21)  (12,22,21,11)
                     (21,11,12)  (21,11,12,22)
                     (21,11,22)  (21,11,22,12)
                     (21,12,11)  (21,12,11,22)
                     (21,12,22)  (21,12,22,11)
                     (21,22,11)  (21,22,11,12)
                     (21,22,12)  (21,22,12,11)
                     (22,11,12)  (22,11,12,21)
                     (22,11,21)  (22,11,21,12)
                     (22,12,11)  (22,12,11,21)
                     (22,12,21)  (22,12,21,11)
                     (22,21,11)  (22,21,11,12)
                     (22,21,12)  (22,21,12,11)

Crossrefs

Cf. A000522, A000595, A002416, A095661, A229865, A326209, A326237, A326251, A326252, A326258.

Programs

  • Mathematica
    Table[Sum[k!*Binomial[n^2,k],{k,0,n^2}],{n,0,4}]

Formula

a(n) = A000522(n^2).

A326278 Number of n-vertex, 2-edge multigraphs that are not nesting. Number of n-vertex, 2-edge multigraphs that are not crossing.

Original entry on oeis.org

0, 0, 1, 9, 34, 90, 195, 371, 644, 1044, 1605, 2365, 3366, 4654, 6279, 8295, 10760, 13736, 17289, 21489, 26410, 32130, 38731, 46299, 54924, 64700, 75725, 88101, 101934, 117334, 134415, 153295, 174096, 196944, 221969, 249305, 279090, 311466, 346579, 384579
Offset: 0

Views

Author

Gus Wiseman, Jun 23 2019

Keywords

Comments

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Examples

			The a(3) = 9 non-crossing multigraphs:
  {12,12}
  {12,13}
  {12,23}
  {13,12}
  {13,13}
  {13,23}
  {23,12}
  {23,13}
  {23,23}

Crossrefs

A326247(n) <= a(n) <= A000537(n).
The case for 2-edge simple graphs (rather than multigraphs) is A117662.
Cf. A000108, A001519, A006125, A016098, A054726, A095661.
Cf. A326210, A326243, A326244, A326248, A326250.

Programs

  • Mathematica
    croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x

Formula

Conjectures from Colin Barker, Jun 25 2019: (Start)
G.f.: x^2*(1 + 4*x - x^2) / (1 - x)^5.
a(n) = (n*(3 - 4*n + n^3)) / 6 .
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)

A377069 Triangle read by rows: T(n,k) is the number of (k+1)-vertex dominating sets of the (n+1)-path graph that include the first vertex.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 1, 5, 4, 1, 0, 0, 5, 9, 5, 1, 0, 0, 3, 13, 14, 6, 1, 0, 0, 1, 13, 26, 20, 7, 1, 0, 0, 0, 9, 35, 45, 27, 8, 1, 0, 0, 0, 4, 35, 75, 71, 35, 9, 1, 0, 0, 0, 1, 26, 96, 140, 105, 44, 10, 1, 0, 0, 0, 0, 14, 96, 216, 238, 148, 54, 11, 1
Offset: 0

Views

Author

Andrew Howroyd, Oct 21 2024

Keywords

Comments

T(n,k) is also the number of (k+1)-vertex dominating sets of the (n+2)-path graph that do not include the first vertex.

Examples

			Triangle begins:
  1;
  1, 1;
  0, 2, 1;
  0, 2, 3,  1;
  0, 1, 5,  4,  1;
  0, 0, 5,  9,  5,  1;
  0, 0, 3, 13, 14,  6,   1;
  0, 0, 1, 13, 26, 20,   7,   1;
  0, 0, 0,  9, 35, 45,  27,   8,  1;
  0, 0, 0,  4, 35, 75,  71,  35,  9,  1;
  0, 0, 0,  1, 26, 96, 140, 105, 44, 10, 1;
  ...
Corresponding to T(4,2) = 5, a path graph with 5 vertices has the following 3-vertex dominating sets that include the first vertex (x marks a vertex in the set):
   x . . x x
   x . x . x
   x . x x .
   x x . . x
   x x . x .

Links

Crossrefs

Row sums are A047081.
Column sums are A008776.
Diagonals include A000012, A000027, A000096, A008778, A095661.
Cf. A169623, A212633.

Programs

  • PARI
    T(n)={[Vecrev(p) | p<-Vec((1 + x)/(1 - y*x - y*x^2 - y*x^3) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

Formula

G.f.: (1 + x)/(1 - y*x - y*x^2 - y*x^3).
A212633(n,k) = T(n-1, k-1) + T(n-2, k-1).
Previous Showing 11-13 of 13 results.

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

Content is available under The OEIS End-User License Agreement.