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A236859 The length of the initial ascent 123... in the n-th Catalan numeral, A239903(n).

%I A236859 #19 May 09 2014 23:14:09
%S A236859 0,1,1,1,2,1,1,1,1,1,2,2,2,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,
%T A236859 2,2,2,3,3,3,3,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A236859 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2
%N A236859 The length of the initial ascent 123... in the n-th Catalan numeral, A239903(n).
%H A236859 Antti Karttunen, <a href="/A236859/b236859.txt">Table of n, a(n) for n = 0..16796</a>
%F A236859 a(0) = 0, and for n>=1, a(n) = A126307(A081291(n))-1.
%F A236859 Each n occurs for the first time (as a record) at the position (C_{n+1})-1, so we have a(A001453(n+1)) = n for all n.
%e A236859 A239903(1) = 1, thus a(1) = 1.
%e A236859 A239903(2) = 10, thus a(2) = 1.
%e A236859 A239903(4) = 12, thus a(4) = 2.
%e A236859 A239903(39) = 1232, thus a(39) = 3.
%e A236859 A239903(58784) = 1234567899, thus a(58784) = 9.
%e A236859 Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 10, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.
%o A236859 (Scheme) (define (A236859 n) (if (zero? n) n (- (A126307 (A081291 n)) 1)))
%Y A236859 Cf. A239903, A236855, A081291, A126307.
%K A236859 nonn
%O A236859 0,5
%A A236859 _Antti Karttunen_, Apr 18 2014