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 A115590 #32 Jan 09 2025 09:50:29
%S A115590 0,1,8,729,389017000,58871587162270593034051001,
%T A115590 204040901322752673844230437877671861543858084850895762746141813554591014612008
%N A115590 a(0) = 0; a(n) = (1+a(n-1))^3 for n > 0.
%C A115590 Take the rooted ternary tree of depth n, with (3^(n+1) - 1) / 2 labeled nodes. Let the number of rooted subtrees be a(n). For example, for n = 1 the a(2) = 8 subtrees are:
%C A115590 R...R...R...R......R.......R...R.......R
%C A115590 .../....|....\..../.\...../|...|\...../|\
%C A115590 ..o.....o.....o..o...o...o.o...o.o...o.o.o
%C A115590 Then a(n+1) = (1+a(n))^3.
%H A115590 Geir Agnarsson, Elie Alhajjar, and Aleyah Dawkins, <a href="https://arxiv.org/abs/2312.11379">On locally finite ordered rooted trees and their rooted subtrees</a>, arXiv:2312.11379 [math.CO], 2023.
%H A115590 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
%H A115590 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%H A115590 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A115590 As for A004019, it follows from Aho and Sloane that there is a constant c such that a(n) is the nearest integer to c^(3^n). In fact a(n) = nearest integer to b^(3^n) - 1 where b = 2.0804006677503193521177452323719035237099784935372250879749088464344434056773788...
%t A115590 {0}~Join~RecurrenceTable[{a[n]==(a[n-1]+1)^3, a[0]==1},a,{n,0,8}] (* _Vaclav Kotesovec_, May 21 2015 *)
%Y A115590 Cf. A000578, A004019, A368326, A368327.
%K A115590 easy,nonn
%O A115590 0,3
%A A115590 _Paolo Bonzini_, Mar 15 2006; corrected Apr 06 2006 and Jan 19 2007
%E A115590 Name edited by _Michael De Vlieger_, Dec 21 2023