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Number of 2-divided words of length n over a 4-letter alphabet.
2

%I #24 May 06 2021 08:11:10

%S 0,0,6,40,186,816,3396,14040,57306,233000,943608,3813000,15378716,

%T 61946640,249260316,1002158880,4026527706,16169288640,64901712996,

%U 260410648680,1044535993800,4188615723280,16792541033556,67309233561240,269746851976156

%N Number of 2-divided words of length n over a 4-letter alphabet.

%C See A210109 for further information.

%C It appears that A027377 gives the number of 2-divided words that have a unique division into two parts. - _David Scambler_, Mar 21 2012

%C From _R. J. Mathar_, Mar 25 2012: (Start)

%C Row sums of the following table which shows how many words of length n over a 4-letter alphabet are 2-divided in k>=1 different ways:

%C 6;

%C 20, 20;

%C 60, 66, 60;

%C 204, 204, 204, 204;

%C 670, 690, 676, 690, 670;

%C 2340, 2340, 2340, 2340, 2340, 2340;

%C 8160, 8220, 8160, 8226, 8160, 8220, 8160;

%C First column of the following triangle which shows how many words of length n over a 4-letter alphabet are k-divided:

%C 6;

%C 40, 4;

%C 186, 60, 1;

%C 816, 374, 44, 0;

%C 3396, 1960, 450, 12, 0;

%C 14040, 9103, 3175, 275, 0, 0;

%C 57306, 40497, 17977, 2915, 66, 0, 0;

%C 233000, 174127, 91326, 22243, 1318,..

%C (End)

%F a(n) = 4^n - A001868(n) (see A209970 for proof).

%Y Cf. A210109, A209970, A001868.

%K nonn,more

%O 1,3

%A _N. J. A. Sloane_, Mar 21 2012

%E a(1)-a(10) computed by _R. J. Mathar_, Mar 20 2012

%E a(13) onwards from _N. J. A. Sloane_, Mar 21 2012