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 A197433 Sum of distinct Catalan numbers: a(n) = Sum_{k>=0} A030308(n,k)*C(k+1) where C(n) is the n-th Catalan number (A000108). (C(0) and C(1) not treated as distinct.) 13
 0, 1, 2, 3, 5, 6, 7, 8, 14, 15, 16, 17, 19, 20, 21, 22, 42, 43, 44, 45, 47, 48, 49, 50, 56, 57, 58, 59, 61, 62, 63, 64, 132, 133, 134, 135, 137, 138, 139, 140, 146, 147, 148, 149, 151, 152, 153, 154, 174, 175, 176, 177, 179, 180, 181, 182, 188, 189, 190, 191, 193, 194, 195, 196 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Replace 2^k with A000108(k+1) in binary expansion of n. From Antti Karttunen, Jun 22 2014: (Start) On the other hand, A244158 is similar, but replaces 10^k with A000108(k+1) in decimal expansion of n. This sequence gives all k such that A014418(k) = A239903(k), which are precisely all nonnegative integers k whose representations in those two number systems contain no digits larger than 1. From this also follows that this is a subsequence of A244155. (End) LINKS Antti Karttunen, Table of n, a(n) for n = 0..8191 FORMULA For all n, A244230(a(n)) = n. - Antti Karttunen, Jul 18 2014 G.f.: (1/(1 - x))*Sum_{k>=0} Catalan number(k+1)*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jul 23 2017 MATHEMATICA nmax = 63; a[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}]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 18 2021, after Ilya Gutkovskiy *) CROSSREFS Characteristic function: A176137. Subsequence of A244155. Cf. A000108, A030308, A197432, A014418, A239903, A244158, A244159, A244230, A244231, A244232, A244315, A244316. Cf. also A060112. Other sequences that are built by replacing 2^k in binary representation with other numbers: A022290 (Fibonacci), A029931 (natural numbers), A059590 (factorials), A089625 (primes), A197354 (odd numbers). Sequence in context: A287339 A039086 A280060 * A153781 A255398 A050025 Adjacent sequences: A197430 A197431 A197432 * A197434 A197435 A197436 KEYWORD easy,nonn AUTHOR Philippe Deléham, Oct 15 2011 EXTENSIONS Name clarified by Antti Karttunen, Jul 18 2014 STATUS approved

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Last modified June 25 04:54 EDT 2024. Contains 373697 sequences. (Running on oeis4.)