A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3.

1, 3, 4, 13, 53, 690, 36571, 25233991, 922832284862, 23286741570717144243, 21489756930695820973683319349467, 500426416062641238759467086706254193219790764168482, 10754042042885415070816603338436200915110904821126871858491675028294447933424899095

Offset: 0

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy
• nice

References

• Archimedeans Problems Drive, Eureka, 19 (1957), 13.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..17
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
• A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
• Index entries for sequences of form a(n+1)=a(n)^2 + ...

Crossrefs

Cf. A001622 (phi), A258112.

Programs

GAP

a:=[1,3];; for n in [3..13] do a[n]:=a[n-1]*a[n-2]+1; od; a; # G. C. Greubel, Sep 19 2019

Haskell

a001056 n = a001056_list !! n
a001056_list = 1 : 3 : (map (+ 1 ) $
               zipWith (*) a001056_list $ tail a001056_list)
-- Reinhard Zumkeller, Aug 15 2012

Magma

I:=[1,3]; [n le 2 select I[n] else Self(n-1)*Self(n-2) + 1: n in [1..13]]; // G. C. Greubel, Sep 19 2019

Maple

a:= proc (n) option remember;
if n=0 then 1
elif n=1 then 3
else a(n-1)*a(n-2) + 1
end if
end proc;
seq(a(n), n = 0..13); # G. C. Greubel, Sep 19 2019

Mathematica

RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]*a[n-2]+1},a,{n,0,14}] (* Harvey P. Dale, Jul 17 2011 *)
t = {1, 3}; Do[AppendTo[t, t[[-1]] * t[[-2]] + 1], {n, 2, 14}] (* T. D. Noe, Jun 25 2012 *)

PARI

m=13; v=concat([1,3], vector(m-2)); for(n=3, m, v[n]=v[n-1]*v[n-2] +1 ); v \\ G. C. Greubel, Sep 19 2019

Sage

def a(n):
    if (n==0): return 1
    elif (n==1): return 3
    else: return a(n-1)*a(n-2) + 1
[a(n) for n in (0..13)] # G. C. Greubel, Sep 19 2019

Formula

a(n) ~ c^(phi^n), where c = A258112 = 1.7978784900091604813559508837..., phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Dec 17 2014