This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A273400 #20 May 27 2025 18:18:23 %S A273400 1,1,2,39044429911904443959240 %N A273400 a(n) = Catalan(Catalan(Catalan(n))). %C A273400 a(4) has 1610164 digits and it is thus too large to be included. %C A273400 Conjecture. The number of digits of a(n) grows asymptotically faster than Catalan(n), i.e., only a finite number of terms of a(n) has the number of digits less than the value of Catalan(n). %C A273400 This also appears to be true for the Fibonacci sequence (A000045) and the sequence of powers of 2 (A000079): it takes two additional iterations of these sequences for the number of digits of these iterated sequences to grow faster than the corresponding original sequences. However, it appears that it takes only one additional iteration of the factorial (A000142) for this to happen. %C A273400 The number of digits of a(n) grows asymptotically faster than Fibonacci(n), but that is already true for Catalan(Catalan(n)) (A273399). %F A273400 a(n) = A000108(A000108(A000108(n))). %e A273400 For n = 2, a(2) = Catalan(Catalan(Catalan(2))) = Catalan(Catalan(2)) = Catalan(2) = 2 as Catalan(2) = 2. %p A273400 a:= ((n-> binomial(2*n, n)/(n+1))@@3): %p A273400 seq(a(n), n=0..3); # _Alois P. Heinz_, May 27 2025 %t A273400 CatalanNumber[CatalanNumber[CatalanNumber[Range[0, 3]]]] %o A273400 (PARI) for(n=0, 3, cn=binomial(2*n, n)/(n+1); cn2=binomial(2*cn, cn)/(cn+1); cn3=binomial(2*cn2, cn2)/(cn2+1); print1(cn3 ", ")) %Y A273400 Cf. A000108 (Catalan), A273399 (Catalan(Catalan(n))), A058051. %K A273400 nonn %O A273400 0,3 %A A273400 _Waldemar Puszkarz_, May 21 2016