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A227185 The largest part in the unordered partition encoded in the runlengths of the binary expansion of n.

%I A227185 #33 Jan 30 2019 19:48:18
%S A227185 0,1,1,2,2,1,2,3,3,2,1,2,3,2,3,4,4,3,2,3,2,1,2,3,4,3,2,3,4,3,4,5,5,4,
%T A227185 3,4,3,2,3,4,3,2,1,2,3,2,3,4,5,4,3,4,3,2,3,4,5,4,3,4,5,4,5,6,6,5,4,5,
%U A227185 4,3,4,5,4,3,2,3,4,3,4,5,4,3,2,3,2,1,2
%N A227185 The largest part in the unordered partition encoded in the runlengths of the binary expansion of n.
%C A227185 The bijective encoding of nonordered partitions via compositions (ordered partitions) present in the binary expansion of n is explained in A227184.
%C A227185 It appears that a(4n+2) = a(2n+1). - _Ralf Stephan_, Jul 20 2013
%H A227185 Antti Karttunen, <a href="/A227185/b227185.txt">Table of n, a(n) for n = 0..8192</a>
%F A227185 Defining formula:
%F A227185 a(0)=0; and for n>=1, a(n) = A029837(n+1) - (A005811(n)-1). [Because the largest part in the unordered partition in this encoding scheme is computed as (c_1 + (c_2-1) + (c_3-1) + ... + (c_k-1)) where c_1 .. c_k are the parts of the k-part composition that sum together as c_1 + c_2 + ... + c_k = A029837(n+1) (the binary width of n), so we subtract from the total binary width of n the number of runs (A005811) minus 1.]
%F A227185 Equivalently: a(n) = A092339(n)+1 for n>0.
%F A227185 a(n) = A005811(A129594(n)). [This just states the fact that when conjugating a partition, the largest part of an old partition will be the number of the parts in the new, conjugated partition.]
%e A227185 12 has binary expansion "1100", for which the lengths of runs (consecutive blocks of 0- or 1-bits) are [2,2]. Converting this to a partition in the manner explained in A227184 gives the partition {2+3}. Its largest part is 3, thus a(12)=3, which is actually the first time when this sequence differs from A043276.
%t A227185 Table[Function[b, Max@ Accumulate@ Prepend[If[Length@ b > 1, Rest[b] - 1, {}], First@ b] - Boole[n == 0]]@ Map[Length, Split@ Reverse@ IntegerDigits[ n, 2]], {n, 0, 120}] // Flatten (* _Michael De Vlieger_, May 09 2017 *)
%o A227185 (Scheme)
%o A227185 (define (A227185 n) (if (zero? n) n (+ 1 (- (A029837 (+ 1 n)) (A005811 n)))))
%o A227185 (define (A227185v2 n) (if (zero? n) n (car (reverse (binexp_to_ascpart n))))) ;; Alternative definition, using the auxiliary functions given in A227184.
%Y A227185 For all n, A005811(n) = a(A129594(n)). Cf. also A136480 (for n>= 1, gives the smallest part) and A227183, A227184, A226062, A092339, A227147.
%Y A227185 a(n) gives the rightmost nonzero term on the n-th row of A227189.
%K A227185 nonn,base
%O A227185 0,4
%A A227185 _Antti Karttunen_, Jul 05 2013