A058181 Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.

1, 0, -1, 1, 2, 3, 7, 46, 2109, 4447835, 19783236185116, 391376433956083065015485621, 153175513056180249189030531428945090978436751221570525

Offset

0,5

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..16

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart. 11 (1973), 429-437.

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fib. Quart. 11 (1973), 429-437. [See here for the missing page.]

Index entries for sequences of form a(n+1) = a(n)^2 + ...

Formula

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

For n >= 4, a(n) = ceiling(c^(2^n)) with c=1.0303497388742578142745024606710866\

16436302563960998408889321488508667424048981473368773165340730475719244472111...

and c^(1/4) = 1.0075025785879710605024343257517358... - Benoit Cloitre, Apr 16 2007

Example

a(6) = a(5)^2 - a(4) = 3^2 - 2 = 7.

Mathematica

Join[{a=1, b=0}, Table[c=b^2-a; a=b; b=c, {n, 13}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
RecurrenceTable[{a[0]==1, a[1]==0, a[n]==a[n-1]^2 - a[n-2]}, a, {n, 13}] (* Vincenzo Librandi, Nov 11 2012 *)

Prog

(PARI) a(n)=if(n<0, a(-1-n), if(n<2, 1-n, a(n-1)^2-a(n-2))) /* Michael Somos, May 05 2005 */
(Magma) I:=[1, 0]; [n le 2 select I[n] else Self(n-1)^2 - Self(n-2): n in [1..15]]; // G. C. Greubel, Jun 09 2019
(Sage)
def a(n):
    if (n==0): return 1
    elif (n==1): return 0
    else: return a(n-1)^2 - a(n-2)
[a(n) for n in (0..15)] # G. C. Greubel, Jun 09 2019
(GAP) a:=[1, 0];; for n in [3..15] do a[n]:=a[n-1]^2-a[n-2]; od; a; # G. C. Greubel, Jun 09 2019

Crossrefs

Cf. A058182.

Sequence in context: A343522 A086542 A349893 * A198959 A000231 A090593

Adjacent sequences: A058178 A058179 A058180 * A058182 A058183 A058184

Keyword

sign

Author

Henry Bottomley, Nov 15 2000

Status

approved