A219024 Number of length n mixed-radix numbers with base [2, 3, 4, ...] (factorial base) such that the parities of adjacent digits differ.

1, 2, 3, 6, 16, 48, 180, 720, 3456, 17280, 100800, 604800, 4147200, 29030400, 228614400, 1828915200, 16257024000, 146313216000, 1448500838400, 14485008384000, 158018273280000, 1738201006080000, 20713561989120000, 248562743869440000, 3212195459235840000

Offset

0,2

Comments

Leading zeros are permitted.

The base [2, 3, 4, ...] in the definition is sometimes called "rising factorial base". Using the "falling factorial base" [..., 4, 3, 2] gives the same sequence.

The number of such factorial numbers without any condition for the digit is (n+1)!.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Wikipedia, Factorial number system.

Formula

For n > 1, a(2n-1) = n * a(2n-2) = (n^2-1) * a(2n-3). - Jon Perry, Nov 15 2012

Example

The a(4) = 16 such numbers are (dots for zeros):
[ 1]   [ . 1 . 1 ]
[ 2]   [ . 1 . 3 ]
[ 3]   [ . 1 2 1 ]
[ 4]   [ . 1 2 3 ]
[ 5]   [ 1 . 1 . ]
[ 6]   [ 1 . 1 2 ]
[ 7]   [ 1 . 1 4 ]
[ 8]   [ 1 . 3 . ]
[ 9]   [ 1 . 3 2 ]
[10]   [ 1 . 3 4 ]
[11]   [ 1 2 1 . ]
[12]   [ 1 2 1 2 ]
[13]   [ 1 2 1 4 ]
[14]   [ 1 2 3 . ]
[15]   [ 1 2 3 2 ]
[16]   [ 1 2 3 4 ]

Maple

a:= proc(n) option remember; `if`(n<4, 1+(5+(n-3)*n)*n/3,
       (2*(n-6)*(n+1) *a(n-1)+ n*(n-1)*(n-2)*(n+3) *a(n-2))/
       (4*(n-3)*(n+2)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 15 2012

Mathematica

a[0] = 1; a[1] = 2; a[n_?EvenQ] = a[n] = n*(n+4)/(2*(n+2))*a[n-1]; a[n_?OddQ] := a[n] = (n+1)/2*a[n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 08 2015 *)

Crossrefs

Sequence in context: A089872 A006402 A284417 * A145860 A305190 A087983

Adjacent sequences: A219021 A219022 A219023 * A219025 A219026 A219027

Keyword

nonn,base

Author

Joerg Arndt, Nov 10 2012

Status

approved