A244230 a(n) is the least k such that A197433(k) >= n.

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, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 24, 24, 24, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 32, 32, 32

Offset: 0

Author

Antti Karttunen, Jun 25 2014

Keywords

• nonn

Comments

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).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..6918

Crossrefs

The first differences give A176137 from its term a(1) onward.

Cf. A000108, A197433.

Programs

Mathematica

nmax = 68;
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}]];
a[n_] := For[k = 0, True, k++, If[A197433[k] >= n, Return[k]]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 18 2021, after Ilya Gutkovskiy in A197433 *)

Formula

For all n >= 0, a(A197433(n)) = n. [This works as an inverse function for the injection A197433].