%I A081088
%S A081088 2,2,10,260,703300,128651592765800,11640481755119007104771565805489000,
%T A081088 17432395734015190050301181934013684788461125501100342391858949624062957005321114000
%N A081088 Smallest partial quotients of a series of continued fractions that have 
               a sum of unity, where the n-th continued fraction term is [0; a(n),
               a(n-1),...,a(1)], so that sum(n>0, [0; a(n),a(n-1),...,a(1)] ) = 
               1, with a(1)=2.
%C A081088 a(n+1) appears to be divisible by a(n) for n>0; a(n+1)/a(n) = A081089(n). 
               Also log(a(n+1))/log(a(n)) -> 1+sqrt(2). The 8-th term has 79 digits, 
               while the 9-th term has 199 digits.
%C A081088 The convergents of the continued fraction series equal the ratios of 
               the terms of A081090: [0; a(n),a(n-1),...,a(1)] = A081090(n)/A080190(n+1) 
               for n>1; thus a(n) = ( A081090(n+1) - A081090(n-1) )/A081090(n) for 
               n>1. - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 05 2003
%F A081088 a(n) = A081090(n)*A081090(n-1) for n>2, where A081090(n)^2 + 1 = a(n+1)/
               a(n).
%e A081088 1 = [0;2] + [0;2,2] + [0;10,2,2] + [0;260,10,2,2] + [0;703300,260,10,
               2,2] + [0;128651592765800,703300,260,10,2,2] +... = .5 + .4 + .0961538461 
               + .0038447319 + .0000014218 + ...
%e A081088 At n=4, [0;260,10,2,2] = A081090(4)/A081090(5) = 52/13525. - Paul D. 
               Hanna (pauldhanna(AT)juno.com), Mar 05 2003
%Y A081088 Cf. A081086, A081089, A081090.
%Y A081088 Sequence in context: A011248 A069240 A000371 this_sequence A001038 A027623 
               A037234
%Y A081088 Adjacent sequences: A081085 A081086 A081087 this_sequence A081089 A081090 
               A081091
%K A081088 cofr,nonn
%O A081088 1,1
%A A081088 Hans Havermann (pxp(AT)rogers.com) and Paul D. Hanna (pauldhanna(AT)juno.com), 
               Mar 05 2003