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%I #61 Jul 03 2020 22:53:16
%S 1,2,4,14,184,33674,1133904604,1285739649838492214,
%T 1653126447166808570252515315100129584,
%U 2732827050322355127169206170438813672515557678636778921646668538491883474
%N Quadratic recurrence from Sylvester's sequence, but starting with a(0)=1 and a(1)=2.
%C a(n) is the size of the set S(n) constructed recursively as follows: Let S(1) = {a,b} and let P(S) be the set of pairs (s,t) where s,t are members of S and s not equal to t. We define S(n+1) as the union of S(n) and P(S(n)). - _David M. Cerna_, Feb 07 2018
%H David M. Cerna, <a href="/A166105/b166105.txt">Table of n, a(n) for n = 0..12</a>
%F Sum_{n>=0} 1/a(n) = 1.82689305142092757947757234878575... (compare with Sum_{n>=0} 1/A000058(n) = 1).
%F a(n) ~ c^(2^n), where c = 1.385089248334672909882206535871311526236739234374149506334120193387331772... . - _Vaclav Kotesovec_, Jan 19 2015
%F Sum_{n>=1} arctan(1/a(n)) = Pi/4. - _Carmine Suriano_, Apr 07 2015
%F a(0)=1, a(n+1) = a(n)*(a(n)-1) + 2. - _Robert FERREOL_, May 05 2020
%F a(n) = A002065(n) + 1 = (A232806(n) + 1)/2. - _Robert FERREOL_, May 31 2020
%p a := proc(n) option remember: if n=0 then 1 elif n=1 then 2 elif n>=2 then procname(n-1)^2 - procname(n-2)^2 + procname(n-2) fi; end:
%p seq(a(n), n = 0..10); # _Muniru A Asiru_, Feb 07 2018
%p a:=1:A:=a : to 10 do a:=a*(a-1)+2 : A:=A,a od:
%p print(A); # _Robert FERREOL_, May 05 2020
%t RecurrenceTable[{a[n]==a[n-1]^2-a[n-2]^2+a[n-2],a[0]==1,a[1]==2}, a, {n,0,10}] (* _Vaclav Kotesovec_, Jan 19 2015 *)
%o (PARI) a(n)=if(n<2,[1,2][n+1],a(n-1)^2-a(n-2)^2+a(n-2));
%o (GAP) a:= [1, 2];; for n in [3..13] do a[n]:= a[n-1]^2 - a[n-2]^2 + a[n-2]; od; a; # _Muniru A Asiru_, Feb 07 2018
%Y Cf. A000058.
%K nonn
%O 0,2
%A _Jaume Oliver Lafont_, Oct 06 2009
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