A165903 a(n) = (a(n-1)^2 + a(n-2)^2 + a(n-1)*a(n-2))/a(n-3) with three initial ones.

1, 1, 1, 3, 13, 217, 16693, 21717363, 2175145909081, 283430597537694797281, 3699017428454717709381715649628841, 6290488320295607125006566146327310005599469877825552723

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..17

Formula

a(n) ~ 1/6 * c^(((1+sqrt(5))/2)^n), where c = 1.902254978346365075882696720546123493664... . - Vaclav Kotesovec, May 06 2015

a(n) = 6*a(n-1)*a(n-2)-a(n-1)-a(n-2)-a(n-3). - Bruno Langlois, Aug 21 2016

Maple

a:= proc(n, k) option remember;
      if n<3 then 1
    else (a(n-1)^2 + a(n-2)^2 + a(n-1)*a(n-2))/a(n-3)
      fi; end:
seq( a(n), n=0..12); # G. C. Greubel, Dec 19 2019

Mathematica

RecurrenceTable[{a[0]==1, a[1]==1, a[2]==1, a[n]==(a[n-1]^2+a[n-2]^2+a[n-1]*a[n-2])/a[n-3]}, a, {n, 0, 10}] (* Vaclav Kotesovec, May 06 2015 *)
nxt[{a_, b_, c_}]:={b, c, (c^2+b^2+b*c)/a}; NestList[nxt, {1, 1, 1}, 10][[All, 1]] (* Harvey P. Dale, Oct 24 2022 *)

Prog

(PARI) a(n)=if(n<3, 1, (a(n-1)^2 +a(n-2)^2 +a(n-1)*a(n-2))/a(n-3))
(Magma) I:=[1, 1, 1]; [n le 3 select I[n] else (Self(n-1)^2 + Self(n-2)^2 + Self(n-1)*Self(n-2))/Self(n-3): n in [1..12]]; // G. C. Greubel, Dec 19 2019
(Sage)
@CachedFunction
def a(n):
    if (n<3): return 1
    else: return (a(n-1)^2+a(n-2)^2+a(n-1)*a(n-2))/a(n-3)
[a(n) for n in (0..12)] # G. C. Greubel, Dec 19 2019
(GAP) a:=[1, 1, 1];; for n in [4..12] do a[n]:= (a[n-1]^2 + a[n-2]^2 + a[n-1]*a[n-2])/a[n-3]; od; a; # G. C. Greubel, Dec 19 2019

Crossrefs

Cf. A064098, A072882, A165896.

Sequence in context: A112093 A085010 A259988 * A100441 A036680 A111431

Adjacent sequences: A165900 A165901 A165902 * A165904 A165905 A165906

Keyword

nonn

Author

Jaume Oliver Lafont, Sep 29 2009

Extensions

"frac" keyword removed by Jaume Oliver Lafont, Oct 13 2009

Status

approved