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%I #20 Nov 18 2021 12:19:00
%S 0,1,2,3,4,4,5,6,7,8,8,8,8,8,8,9,10,11,12,12,13,14,15,16,16,16,16,16,
%T 16,16,16,16,16,16,16,16,16,16,16,16,16,16,16,17,18,19,20,20,21,22,23,
%U 24,24,24,24,24,24,25,26,27,28,28,29,30,31,32,32,32,32
%N a(n) is the least k such that A197433(k) >= n.
%C For n >= 1, a(n) is the total number of ways the natural numbers in range 1 .. n can be represented as sums of distinct Catalan numbers (A000108). Note that for any one number, number of such solutions may be at most one. In other words, this sequence is one less than the partial sums of A176137 (number of partitions of n into distinct Catalan numbers).
%H Antti Karttunen, <a href="/A244230/b244230.txt">Table of n, a(n) for n = 0..6918</a>
%F For all n >= 0, a(A197433(n)) = n. [This works as an inverse function for the injection A197433].
%t nmax = 68;
%t A197433[n_] := If[n == 0, 0, SeriesCoefficient[(1/(1-x))*Sum[ CatalanNumber[k+1]*x^(2^k)/(1+x^(2^k)), {k, 0, Log[2, n] // Ceiling}], {x, 0, n}]];
%t a[n_] := For[k = 0, True, k++, If[A197433[k] >= n, Return[k]]];
%t Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Nov 18 2021, after _Ilya Gutkovskiy_ in A197433 *)
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A244230 (LEAST-GTE-I 0 0 A197433))
%Y The first differences give A176137 from its term a(1) onward.
%Y Cf. A000108, A197433.
%K nonn
%O 0,3
%A _Antti Karttunen_, Jun 25 2014
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