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A055187 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 1.

%I A055187 #18 Nov 10 2022 16:12:34
%S A055187 1,1,1,3,1,4,1,1,3,6,1,2,3,1,4,8,1,1,2,3,3,2,4,1,6,11,1,3,2,5,3,3,4,2,
%T A055187 6,1,8,13,1,5,2,8,3,4,4,1,5,3,6,2,8,1,11,16,1,7,2,10,3,6,4,3,5,4,6,4,
%U A055187 8,2,11,1,13,18,1,9,2,12,3,9,4,4,5,6,6,1,7,5,8,1,10,3,11,2,13,1,16,22,1
%N A055187 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 1.
%C A055187 Conjectures: limit as n goes to infinity of max {a(k) : 1<=k<=n}/sqrt(n) = 2;
%C A055187 -3 < a(n) - 2*sqrt(n) < 3 for all n;
%C A055187 there are infinitely many n such that a(n)=a(n+1). - _Benoit Cloitre_, Jan 30 2003
%C A055187 After starting with 1, successive segments are generated in adjective-before-noun pairs as in A055186 (i.e., the noun-integers are in increasing order).  See A217780 for the sequence originally placed here, in which the noun-integers are in order of 1st occurrence. - _Clark Kimberling_, Mar 24 2013
%H A055187 Seiichi Manyama, <a href="/A055187/b055187.txt">Table of n, a(n) for n = 1..10000</a>
%e A055187 After writing 1, pairs, written vertically, are as shown:
%e A055187 1..1..3..4 1..6 2 1..8 1 3 2 1..
%e A055187 ...1..1..1 3..1 3 4..1 2 3 4 6..
%t A055187 s = {1}; Do[s = Flatten[{s, {Count[s, #], #} & /@ Union[s]}], {14}]; s  (* A055187 *) (* _Peter J. C. Moses_, Mar 21 2013 *)
%Y A055187 Cf. A055186, A217780.
%K A055187 nonn
%O A055187 1,4
%A A055187 _Clark Kimberling_, Apr 27 2000
%E A055187 Corrected and extended by _Benoit Cloitre_, Jan 30 2003