%I M1533 #60 Aug 03 2024 01:49:27

%S 1,2,5,20,230,26795,359026205,64449908476890320,

%T 2076895351339769460477611370186680,

%U 2156747150208372213435450937462082366919951682912789656986079991220

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

%D M. Abert and P. Diaconis, paper in preparation, 2002.

%D D. Parisse, The Tower of Hanoi and the Stern-Brocot-Array, Thesis, Munich, 1997.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Michael De Vlieger, <a href="/A006893/b006893.txt">Table of n, a(n) for n = 1..13</a>

%H 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 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 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 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F a(n) = A007501(n-1) - 1.

%F a(n+1) = a(n)*(a(n)+3)/2, a(1)=1.

%F 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 a(n) ~ 2 * c^(2^n), where c = 1.16007248510653786919452141287945841802404855231102953089... . - _Vaclav Kotesovec_, Dec 17 2014

%p A006893 := proc(n) option remember; if n=1 then 1 else A006893(n-1)*(A006893(n-1)+3)/2; fi; end;

%t RecurrenceTable[{a[1] == 1, a[n] == a[n-1]*(a[n-1] + 3)/2}, a[n], {n, 10}] (* _Vaclav Kotesovec_, Dec 17 2014 *)

%o (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 Where records occur in A057945, n >= 1.

%Y Cf. A007501.

%K nonn,easy

%O 1,2

%A _Jeffrey Shallit_

%E Additional description from _Andreas M. Hinz_ and _Daniele Parisse_