A166105 Quadratic recurrence from Sylvester's sequence, but starting with a(0)=1 and a(1)=2.

1, 2, 4, 14, 184, 33674, 1133904604, 1285739649838492214, 1653126447166808570252515315100129584, 2732827050322355127169206170438813672515557678636778921646668538491883474

Offset: 0

Author

Jaume Oliver Lafont, Oct 06 2009

Keywords

• nonn

Comments

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

Links

• David M. Cerna, Table of n, a(n) for n = 0..12

Crossrefs

Cf. A000058.

Programs

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

Maple

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:
seq(a(n), n = 0..10); # Muniru A Asiru, Feb 07 2018
a:=1:A:=a : to 10 do a:=a*(a-1)+2 : A:=A,a od:
print(A); # Robert FERREOL, May 05 2020

Mathematica

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 *)

PARI

a(n)=if(n<2,[1,2][n+1],a(n-1)^2-a(n-2)^2+a(n-2));

Formula

Sum_{n>=0} 1/a(n) = 1.82689305142092757947757234878575... (compare with Sum_{n>=0} 1/A000058(n) = 1).

a(n) ~ c^(2^n), where c = 1.385089248334672909882206535871311526236739234374149506334120193387331772... . - Vaclav Kotesovec, Jan 19 2015

Sum_{n>=1} arctan(1/a(n)) = Pi/4. - Carmine Suriano, Apr 07 2015

a(0)=1, a(n+1) = a(n)*(a(n)-1) + 2. - Robert FERREOL, May 05 2020

a(n) = A002065(n) + 1 = (A232806(n) + 1)/2. - Robert FERREOL, May 31 2020