A284604 Quadratic recurrence: a(n+2) = a(n+1)^2 + a(n)^2 + 1, with a(0) = a(1) = 1.

1, 1, 3, 11, 131, 17283, 298719251, 89233191216703091, 7962562414716697755180182566955283, 63402400208259008611705446682872670539115181497111590988296570564371

Offset: 0

Author

Emanuele Munarini, Mar 30 2017

Keywords

• nonn

Crossrefs

Cf. A000283.

Programs

Magma

[n le 2 select 1 else Self(n-1)^2+Self(n-2)^2+1: n in [1..10]]; // Bruno Berselli, Mar 30 2017

Mathematica

RecurrenceTable[{a[n + 2] == a[n + 1]^2 + a[n]^2 + 1, a[0] == 1, a[1] == 1}, a, {n, 0, 12}]
nxt[{a_,b_}]:={b,a^2+b^2+1}; NestList[nxt,{1,1},10][[;;,1]] (* Harvey P. Dale, Feb 16 2025 *)

Maxima

a(n) := if (n=0 or n=1) then 1 else a(n-1)^2 + a(n-2)^2 + 1; makelist(a(n), n, 0, 12);

Formula

a(n+3) = a(n+2)^2 + a(n+2) - a(n)^2, with a(0) = a(1) = 1, a(2) = 3.

a(n) ~ c^(2^n), where c = 1.356519333072951374233963037913978335267300244021120535099185060013... - Vaclav Kotesovec, Apr 15 2017