A000278 a(n) = a(n-1) + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.

0, 1, 1, 2, 3, 7, 16, 65, 321, 4546, 107587, 20773703, 11595736272, 431558332068481, 134461531248108526465, 186242594112190847520182173826, 18079903385772308300945867582153787570051, 34686303861638264961101080464895364211215702792496667048327

Offset: 0

Author

Stephen J. Greenfield (greenfie(AT)math.rutgers.edu)

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 0..25
• W. Duke, Stephen J. Greenfield, and Eugene R. Speer, Properties of a Quadratic Fibonacci Recurrence, J. Integer Seq. 1 (1998), Article #98.1.8.

Crossrefs

Cf. A000283, A058182.

Programs

Magma

[n le 2 select n-1 else Self(n-1) + Self(n-2)^2: n in [1..18]]; // Vincenzo Librandi, Dec 17 2015

Maple

A000278 := proc(n) option remember; if n <= 1 then n else A000278(n-2)^2+A000278(n-1); fi; end;

Mathematica

Join[{a=0,b=1},Table[c=a^2+b;a=b;b=c,{n,16}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
RecurrenceTable[{a[n +2] == a[n +1] + a[n]^2, a[0] == 1, a[1] == 1}, a, {n, 0, 16}] (* Robert G. Wilson v, Apr 14 2017 *)

PARI

a(n)=if(n<2,n>0,a(n-1)+a(n-2)^2)

Sage

def A000278():
    x, y = 0, 1
    while True:
        yield x
        x, y = x + y, x * x
a = A000278(); [next(a) for i in range(18)]  # Peter Luschny, Dec 17 2015

Formula

a(2n) is asymptotic to A^(sqrt(2)^(2n-1)) where A=1.668751581493687393311628852632911281060730869124873165099170786836201970866312366402366761987... and a(2n+1) to B^(sqrt(2)^(2n)) where B=1.693060557587684004961387955790151505861127759176717820241560622552858106116817244440438308887... See reference for proof. - Benoit Cloitre, May 03 2003

Extensions

Name edited by Petros Hadjicostas, Nov 03 2019