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%I #32 Dec 03 2023 09:12:08
%S 2,7,57,3307,10939557,119673918295807,14321846720271609085072077057,
%T 205115293478954645768397227034180943592279329877217858307,
%U 42072283618957694230389567430137958296609066493047345973782287300661413651741392431587718724877522268597146764557
%N a(n) = 1 + a(n-1)*(a(n-1) + 1), with a(0)=2.
%C For n > 1, a(n) has digital root 3 or 4 depending on whether n is odd or even.
%C The 5-adic valuation of a(n)^2 + 1 is n+1. - _William Hu_, Dec 01 2023
%D T. Koshy, "Intriguing Properties Of Three Related Number Sequences", in Journal of Recreational Mathematics, vol. 32(3) 210-3 2003-4 Baywood NY.
%F From _Gerald McGarvey_, Dec 12 2007: (Start)
%F For n > 0, a(n) = Sum_{k=0..n-1} a(k)^2 + n + 2.
%F Conjecture: a(n) is asymptotic to d - 1/2 -(5/2^3)/d -(65/2^7)/d^3 -(650/2^11)/d^5 -(19045/2^15)/d^7 -(274950/2^19)/d^9 -(6979050/2^23)/d^11 -(130292500/2^27)/d^13 ... where d = c^(2^n) and c is a constant = 1.288203192684485177845610784851700404829443712770079185959554466777577486352420255603915828361833141546.... (End)
%p a[0]:=2: for n from 1 to 8 do a[n]:=1+a[n-1]*(a[n-1]+1) od: seq(a[n],n=0..8); # _Emeric Deutsch_, Jun 13 2005
%t NestList[1+#(#+1)&,2,10] (* _Harvey P. Dale_, Mar 30 2016 *)
%Y Cf. A005267.
%K nonn
%O 0,1
%A _Lekraj Beedassy_, Apr 11 2005
%E More terms from _Emeric Deutsch_, Jun 13 2005
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