A338614 -id:A338614 - cp's OEIS frontend

A333833 Number of permutations p of [n] such that |p(i) - p(i-1)| <= 2 and |p(i) - p(i-2)| <= 3.

1, 1, 2, 6, 12, 14, 18, 28, 42, 56, 74, 102, 144, 200, 274, 376, 520, 720, 994, 1370, 1890, 2610, 3604, 4974, 6864, 9474, 13078, 18052, 24916, 34390, 47468, 65520, 90436, 124826, 172294, 237814, 328250, 453076, 625370, 863184, 1191434, 1644510, 2269880, 3133064

Offset: 0

Alois P. Heinz, Apr 07 2020

			a(5) = 14: 12345, 12354, 12435, 12453, 13245, 21345, 31245, 35421, 45321, 53421, 54213, 54231, 54312, 54321.
a(6) = 18: 123456, 123465, 123546, 123564, 124356, 132456, 213456, 213465, 312456, 465321, 564312, 564321, 645321, 653421, 654213, 654231, 654312, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..7141
Alois P. Heinz, Animation of a(10) = 74 permutations
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Cf. A003274, A086106, A174700, A263690, A263696, A302510, A307269, A328648, A338614.

Join[{1, 1, 2, 6, 12}, LinearRecurrence[{1, 0, 0, 1}, {14, 18, 28, 42}, 40]] (* Jean-François Alcover, Oct 26 2021 *)

G.f.: -(2*x^8+4*x^7+2*x^6+x^5+5*x^4+4*x^3+x^2+1)/(x^4+x-1).
a(n) = 2*A302510(n-2) for n >= 6.
Limit_{n-> infinity} a(n+1)/a(n) = A086106.

A338738 Number of permutations p of [n] such that |p(i) - p(i-1)| <= |p(i+1) - p(i)|.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 10, 14, 20, 24, 32, 40, 48, 70, 94, 126, 162, 228, 292, 386, 528, 710, 956, 1298, 1730, 2342, 3178, 4192, 5684, 7720, 10340, 14002, 18816, 25372, 34054

Offset: 0

Alois P. Heinz, Nov 05 2020

			a(4) = 6: 1234, 2314, 2341, 3214, 3241, 4321.

Wikipedia, Permutation

Cf. A338614.

b:= proc(s, x, y) option remember; `if`(s={}, 1, add(
      `if`(x=0 or y=0 or abs(x-y)<=abs(y-j),
         b(s minus {j}, y, j), 0), j=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..16);

b[s_, x_, y_] := b[s, x, y] = If[s == {}, 1, Sum[If[x == 0 || y == 0 || Abs[x - y] <= Abs[y - j], b[s ~Complement~ {j}, y, j], 0], {j, s}]];
a[n_] := b[Range[n], 0, 0];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 08 2021, after Alois P. Heinz *)

a(27)-a(34) from Bert Dobbelaere, Nov 15 2020

cp's OEIS Frontend

A333833 Number of permutations p of [n] such that |p(i) - p(i-1)| <= 2 and |p(i) - p(i-2)| <= 3.

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