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 1-6 of 6 results.

A317130 Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 7, 24, 93, 483, 2832, 17515, 123226, 978405, 8312802, 75966887, 756376739, 8070649675, 91320842018, 1099612368110, 14054043139523, 189320856378432, 2682416347625463, 39945105092501742, 623240458310527252, 10160826473676346731, 172871969109661492526
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Examples

			a(2) = 1: 21.
a(3) = 2: 123, 321.
a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.
a(5) = 24: 12543, 13542, 14532, 21354, 21453, 23541, 24531, 31254, 31452, 32145, 32451, 34521, 41253, 41352, 42135, 42351, 43125, 51243, 51342, 52134, 52341, 53124, 54123, 54321.

Links

Crossrefs

Cf. A000217, A097597, A193374, A317111, A317128, A317129, A317131, A317132, A317446.

Programs

  • Maple
    g:= n-> `if`(issqr(8*n+1), 1, 0):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := If[IntegerQ @ Sqrt[8n+1], 1, 0];
b[u_, o_, t_] := b[u, o, t] = If[u+o==0, g[t], If[g[t]==1, Sum[b[u-j, o+j-1, 1], {j, 1, u}], 0] + Sum[b[u+j-1, o-j, t+1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

A273001 Number of permutations of [n] whose cycle lengths are Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 420, 2220, 19020, 130860, 1096920, 9862920, 83843640, 1411202520, 16144792560, 203091829200, 2989264122000, 37012939750800, 597962683188000, 8681244913692000, 126467701221607200, 5006833609034743200, 95602098255580238400
Offset: 0

Views

Author

Alois P. Heinz, May 12 2016

Keywords

Links

Crossrefs

Cf. A000045, A193374, A205801, A218002, A272603, A273994, A317128.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(issqr(5*j^2+4) or issqr(5*j^2-4),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[5*j^2+4] || IntegerQ @ Sqrt[5*j^2-4], a[n-j]*(j-1)!*Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[ a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 30 2017, translated from Maple *)

Formula

E.g.f.: exp(Sum_{n>=2} x^F(n)/F(n)) with F = A000045.

A317129 Number of permutations of [n] whose lengths of increasing runs are squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 40, 151, 571, 2897, 19730, 140190, 953064, 6708323, 54631552, 510143776, 4987278692, 49168919669, 505209884549, 5638095015594, 67921924172174, 852861260421398, 10992380368532792, 147296144926635359, 2082906807168675698, 30973237281668975230
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Examples

			a(3) = 1: 321.
a(4) = 2: 1234, 4321.
a(5) = 9: 12354, 12453, 13452, 21345, 23451, 31245, 41235, 51234, 54321.

Links

Crossrefs

Cf. A000290, A097597, A205801, A317111, A317128, A317130, A317131, A317132, A317445.

Programs

  • Maple
    g:= n-> `if`(issqr(n), 1, 0):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := If[IntegerQ@Sqrt[n], 1, 0];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)

A317131 Number of permutations of [n] whose lengths of increasing runs are prime numbers.

Original entry on oeis.org

1, 0, 1, 1, 5, 19, 80, 520, 2898, 22486, 171460, 1509534, 14446457, 147241144, 1650934446, 19494460567, 248182635904, 3340565727176, 47659710452780, 718389090777485, 11381176852445592, 189580213656445309, 3305258537062221020, 60273557241570401742
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Examples

			a(2) = 1: 12.
a(3) = 1: 123.
a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 19: 12345, 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 23145, 23415, 23514, 24135, 24513, 25134, 34125, 34512, 35124, 45123.

Links

Crossrefs

Cf. A000040, A097597, A218002, A317111, A317128, A317129, A317130, A317132, A317447.

Programs

  • Maple
    g:= n-> `if`(n=0 or isprime(n), 1, 0):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := If[n == 0 || PrimeQ[n], 1, 0];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)
  • Python
    from functools import lru_cache
from sympy import isprime
def g(n): return int(n == 0 or isprime(n))
@lru_cache(maxsize=None)
def b(u, o, t):
  if u + o == 0: return g(t)
  return (sum(b(u-j,  o+j-1,  1) for j in range(1, u+1)) if g(t) else 0) +\
          sum(b(u+j-1, o-j, t+1) for j in range(1, o+1))
def a(n): return b(n, 0, 0)
print([a(n) for n in range(28)]) # Michael S. Branicky, Mar 29 2021 after Alois P. Heinz

A317132 Number of permutations of [n] whose lengths of increasing runs are factorials.

Original entry on oeis.org

1, 1, 2, 5, 17, 70, 350, 2029, 13495, 100813, 837647, 7652306, 76282541, 823684964, 9578815164, 119346454671, 1586149739684, 22397700381817, 334879465463998, 5285103821004717, 87800206978975107, 1531533620821692217, 27987305231654121046, 534688325008397289484
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2018

Keywords

Links

Crossrefs

Cf. A000142, A097597, A272603, A317111, A317128, A317129, A317130, A317131, A317448.

Programs

  • Maple
    g:= proc(n) local i; 1; for i from 2 do
      if n=% then 1; break elif n<% then 0; break fi;
      %*i od; g(n):=%
    end:
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 1), j=1..n)):
seq(a(n), n=0..27);
  • Mathematica
    g[n_] := g[n] = Module[{i, k = 1}, For[i = 2, True, i++,
     If[n == k, k = 1; Break[]]; If[n < k, k = 0; Break[]];
     k = k*i]; k];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021~, after Alois P. Heinz *)

Formula

a(3) = 5: 132, 213, 231, 312, 321.
a(4) = 17: 1324, 1423, 1432, 2143, 2314, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321.

A317444 Number of permutations of [n] whose lengths of increasing runs are distinct Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 5, 6, 19, 212, 40, 757, 2170, 13546, 379084, 8978, 73195, 2702092, 772852, 38833826, 213557110, 2390871412, 150689939006, 9394670, 634504029, 4522073096, 63395566566, 5160905755362, 192831696582, 3068824154606, 289158899744046, 116561588867106
Offset: 0

Views

Author

Alois P. Heinz, Jul 28 2018

Keywords

Links

Crossrefs

Cf. A000045, A317128, A317445, A317446, A317447, A317448.

Programs

  • Maple
    g:= (n, s)-> `if`(n in s or not
    (issqr(5*n^2+4) or issqr(5*n^2-4)), 0, 1):
b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
      `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
       , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
    end:
a:= n-> b(n, 0$2, {}):
seq(a(n), n=0..30);
  • Mathematica
    g[n_, s_] := If[MemberQ[s, n] || !(
     IntegerQ@Sqrt[5*n^2 + 4] || IntegerQ@Sqrt[5*n^2 - 4]), 0, 1];
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
     If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
     {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
a[n_] := b[n, 0, 0, {}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)
Showing 1-6 of 6 results.

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