A210323 Number of 2-divided words of length n over a 3-letter alphabet.

0, 3, 16, 57, 192, 599, 1872, 5727, 17488, 53115, 161040, 487073, 1471680, 4441167, 13392272, 40355877, 121543680, 365895947, 1101089808, 3312442185, 9962240928, 29954639751, 90049997136, 270661616363, 813397065024, 2444101696683, 7343167947040, 22059763982001, 66263812628160

Offset: 1

Author

N. J. A. Sloane, Mar 20 2012

Keywords

• nonn

Comments

See A210109 for further information.

It appears that A027376 gives the number of 2-divided words that have a unique division into two parts. - David Scambler, Mar 21 2012

Row sums of the following irregular triangle W(n,k) which shows how many words of length n over a 3-letter alphabet are 2-divided in k>=1 different ways (3-letter analog of A209919):

3;

8, 8;

18, 21, 18;

48, 48, 48, 48;

116, 124, 119, 124, 116;

312, 312, 312, 312, 312, 312;

810, 828, 810, 831, 810, 828, 810;

2184, 2184, 2192, 2184, 2184, 2192, 2184, 2184;

5880, 5928, 5880, 5928, 5883, 5928, 5880, 5928, 5880;

First column of the following triangle D(n,k) which shows how many words of length n over a 3-letter alphabet are k-divided:

3;

16, 1;

57, 16, 0;

192, 78, 6, 0;

599, 324, 56, 0, 0;

1872, 1141, 343, 15, 0, 0;

5727, 3885, 1534, 166, 0, 0, 0;

17488, 12630, 6067, 1135, 20, 0, 0, 0;

53115, 40315, 22162, 5865, 351, 0, 0, 0, 0;

161040, 126604, ...

- R. J. Mathar, Mar 25 2012

Speculation: W(2n+1,2)=W(2n+1,1) and W(2n,2) = W(2n,1)+W(n,1). W(3n+1,3)=W(3n+1,1); W(3n+2,3)=W(3n+1,1); W(3n,3) = W(3n,1)+W(n,1). - R. J. Mathar, Mar 27 2012

Crossrefs

Cf. A210109, A209970, A001867.

Formula

a(n) = 3^n - A001867(n) (see A209970 for proof).

Extensions

a(1)-a(12) computed by David Scambler, Mar 19 2012; a(13) onwards from N. J. A. Sloane, Mar 20 2012