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Intertwining numbers.
(Formerly M1988)
3

%I M1988 #34 Apr 03 2018 06:26:33

%S 1,1,2,10,148,7384,1380960,1058349286

%N Intertwining numbers.

%C a(4)=10 for example is the number of ways of arranging 1 a, 2 b's, 3 c's and 4 d's so that if we look at any two letters, i and j say, with i<j, then any pair of i's are separated and surrounded by at least one j.

%C If the condition is imposed only on pairs of consecutive letters, we get A003121.

%D C. L. Mallows, Letter to _N. J. A. Sloane_, Nov 11 1980

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Georg Fischer, <a href="/A213457/a213457.txt">Perl program</a>

%e The 10 sequences for n=4 are dcbadcbdcd dcbadcdbcd dcbdacbdcd dcbdacdbcd dcbdcabdcd dcbdcadbcd dcbdcdabcd dcdbacdbcd dcdbcadbcd dcdbcdabcd.

%e For example in dcbdacdbcd we see

%e ..ba...b..

%e .cb..c.bc.

%e d.bd..db.d

%e dc.d.cd.cd

%Y Cf. A003121.

%K nonn,more

%O 1,3

%A _N. J. A. Sloane_. Entry revised and given a new A-number by _N. J. A. Sloane_, Jun 13 2012. The old entry was A004065.

%E a(7) from _David W. Wilson_, Dec 11 1999

%E Definition clarified by _David Applegate_, Jun 14 2012

%E a(8) from _Georg Fischer_, Mar 21 2018