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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A227883 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up.

Original entry on oeis.org

0, 0, 0, 0, 5, 50, 328, 2154, 16751, 144840, 1314149, 12735722, 134159743, 1519210786, 18272249418, 233231701166, 3159471128588, 45243728569842, 682183513506619, 10807962134238068, 179606706777512992, 3123700853586733882, 56737351453843424893
Offset: 0

Views

Author

Alois P. Heinz, Oct 25 2013

Keywords

Examples

			a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 50: 12435, 12534, 13245, ..., 52314, 52413, 53412.

Links

Crossrefs

Column k=1 of A227884.
Cf. A177477, A245758.

Programs

  • Maple
    b:= proc(u, o, t) option remember;
      `if`(t=7, 0, `if`(u+o=0, `if`(t in [4, 5, 6], 1, 0),
      add(b(u-j, o+j-1, [1, 3, 1, 5, 6, 6][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 2, 4, 4, 7, 4][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] =
    If[t == 7, 0, If[u + o == 0, If[4 <= t <= 6, 1, 0],
    Sum[b[u - j, o + j - 1, {1, 3, 1, 5, 6, 6}[[t]]], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, {2, 2, 4, 4, 7, 4}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n! * n, where d = A245758 = 0.782704180171521701844707..., c = 0.575076701401064911213333442496869737011... . - Vaclav Kotesovec, Aug 22 2014

A232899 Number of permutations of [n] cyclically avoiding the consecutive step pattern UDU (U=up, D=down).

Original entry on oeis.org

1, 1, 0, 3, 12, 35, 144, 910, 5976, 39942, 306570, 2698223, 25536132, 257563618, 2813856192, 33154390275, 415692891552, 5523237345701, 77778820305558, 1157352664763569, 18120617730892800, 297774609082108662, 5127157782095091402, 92308888110570124310
Offset: 0

Views

Author

Alois P. Heinz, Dec 02 2013

Keywords

Examples

			a(2) = 0 because 12 and 21 do not avoid UDU (the two U's overlap).
a(3) = 3: 132, 213, 321.
a(4) = 12: 1243, 1342, 1432, 2134, 2143, 2431, 3124, 3214, 3421, 4213, 4312, 4321.
a(5) = 35: 12354, 12453, 12543, ..., 54213, 54312, 54321.

Links

Crossrefs

Column k=0 of A232933.
Cf. A177477, A245758.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t=4, 0,
     `if`(u+o=0, `if`(t=2, 0, 1),
      add(b(u+j-1, o-j, [2, 2, 4][t]), j=1..o)+
      add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)))
    end:
a:= n-> `if`(n<2, 1, n*b(0, n-1, 1)):
seq(a(n), n=0..30);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t == 4, 0,
    If[u + o == 0, If[t == 2, 0, 1],
    Sum[b[u + j - 1, o - j, {2, 2, 4}[[t]]], {j, 1, o}] +
    Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}]]];
a[n_] := If[n < 2, 1, n b[0, n - 1, 1]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Formula

a(n) ~ d^n * n!, where d = A245758 = 0.782704180171521701844707497734609... . - Vaclav Kotesovec, Aug 22 2014

A245758 Decimal expansion of a constant related to A232899.

Original entry on oeis.org

7, 8, 2, 7, 0, 4, 1, 8, 0, 1, 7, 1, 5, 2, 1, 7, 0, 1, 8, 4, 4, 7, 0, 7, 4, 9, 7, 7, 3, 4, 6, 0, 9, 0, 5, 5, 0, 2, 1, 3, 1, 2, 9, 5, 0, 9, 4, 8, 6, 3, 7, 5, 1, 4, 7, 7, 5, 8, 3, 1, 8, 5, 2, 0, 8, 6, 5, 0, 8, 9, 7, 3, 8, 9, 0, 8, 8, 4, 7
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 22 2014

Keywords

Examples

			0.78270418017152170184470749773460905502131295094863751477583185208650897389...

Crossrefs

Cf. A232899, A177477, A227883.

Formula

Equals lim n -> infinity (A232899(n)/n!)^(1/n).
Equals lim n -> infinity (A177477(n)/n!)^(1/n).
Equals lim n -> infinity (A227883(n)/n!)^(1/n).
Previous Showing 11-13 of 13 results.

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