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A006893 Smallest number whose representation requires n triangular numbers with greedy algorithm; also number of 1-2 rooted trees of height n.

%I A006893 M1533 #60 Aug 03 2024 01:49:27
%S A006893 1,2,5,20,230,26795,359026205,64449908476890320,
%T A006893 2076895351339769460477611370186680,
%U A006893 2156747150208372213435450937462082366919951682912789656986079991220
%N A006893 Smallest number whose representation requires n triangular numbers with greedy algorithm; also number of 1-2 rooted trees of height n.
%D A006893 M. Abert and P. Diaconis, paper in preparation, 2002.
%D A006893 D. Parisse, The Tower of Hanoi and the Stern-Brocot-Array, Thesis, Munich, 1997.
%D A006893 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006893 Michael De Vlieger, <a href="/A006893/b006893.txt">Table of n, a(n) for n = 1..13</a>
%H A006893 Mee Seong Im and Can Ozan Oğuz, <a href="https://www.researchgate.net/publication/351823558_Natural_transformations_between_induction_and_restriction_on_iterated_wreath_product_of_symmetric_group_of_order_2">Natural transformations between induction and restriction on iterated wreath product of symmetric group of order 2</a>, (2021).
%H A006893 E. Lemoine, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k2011936/f75.image">Note sur deux nouvelles décompositions des nombres entiers</a>, Assoc. française pour l'avancement des sciences. Vol. 29, Tome 2, pp. 72-74, 1900.
%H A006893 Sridhar Narayanan, <a href="https://arxiv.org/abs/1712.02507">The Representation Theory of 2-Sylow Subgroups of the Symmetric Group</a>, arXiv:1712.02507 [math.RT], 2017.
%H A006893 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%H A006893 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A006893 a(n) = A007501(n-1) - 1.
%F A006893 a(n+1) = a(n)*(a(n)+3)/2, a(1)=1.
%F A006893 a(0) = 1, a(n) = Sum_{i=0..n-1} t(a(i)), where t(n)=n*(n+1)/2. - _Jon Perry_, Feb 14 2004
%F A006893 a(n) ~ 2 * c^(2^n), where c = 1.16007248510653786919452141287945841802404855231102953089... . - _Vaclav Kotesovec_, Dec 17 2014
%p A006893 A006893 := proc(n) option remember; if n=1 then 1 else A006893(n-1)*(A006893(n-1)+3)/2; fi; end;
%t A006893 RecurrenceTable[{a[1] == 1, a[n] == a[n-1]*(a[n-1] + 3)/2}, a[n], {n, 10}] (* _Vaclav Kotesovec_, Dec 17 2014 *)
%o A006893 (PARI) a=vector(20); a[1]=1; for(n=2, #a, a[n]=a[n-1]*(a[n-1]+3)/2); a \\ _Altug Alkan_, Apr 04 2018
%Y A006893 Where records occur in A057945, n >= 1.
%Y A006893 Cf. A007501.
%K A006893 nonn,easy
%O A006893 1,2
%A A006893 _Jeffrey Shallit_
%E A006893 Additional description from _Andreas M. Hinz_ and _Daniele Parisse_