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%I #7 Dec 05 2013 19:55:58
%S 1,212,132323,1234243434,123234535454545,123234345464656565656,
%T 1232343454545657576767676767,123234345454565656767868687878787878,
%U 123234345454565656767676787897979898989898989
%N Smallest number having k digits "k" (k=1,...,n) but any two adjacent digits are different.
%C How many such numbers can be formed?
%C The first (n-1)(n-2)/2 digits of a(n) (part (1) of the formula) remain the same for all subsequent terms. (M. F. Hasler, Jun 22 2007)
%C Terms of the sequence do not really depend on the base: for any base b>n, terms a(1)..a(n) would read the same. Thus one could add "(in base n+1)" to the definition and delete the keyword "base". (M. F. Hasler, Jun 22 2007)
%C The sequence could be extended to terms n>9 in two ways: (1) by writing a(n) according to the prescription in base n+1, but recording the corresponding value in base 10; (2) by providing a convention for encoding digits d > 9, e.g. by pre-pending them with '0#' (knowing they will never occur at the beginning of a(n)), where # is the number of characters used to write the digit (encoded recursively in the same way if #>9). (M. F. Hasler, Jun 22 2007)
%F To get a(n): (1) start with an empty string and always concatenate the smallest possible of the remaining digits, until there are 2n-1 digits left (n "n"s and n-1 other digits); (2) insert the n-1 other digits in-between the "n"s and concatenate this result to the first string. (M. F. Hasler, Jun 22 2007)
%e a(3) = 132323 using 1, 2,2 and 3,3,3 with no two adjacent numbers same.
%K base,nonn
%O 1,2
%A _Amarnath Murthy_, Feb 05 2003
%E Corrected and extended by _M. F. Hasler_, Jun 22 2007
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