A001056 a(n) = a(n-1)*a(n-2) + 1, a(0) = 1, a(1) = 3. (Formerly M2378 N0944)

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

Offset

0,2

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, alternative link.

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

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A001622 (phi), A258112.

Sequence in context: A062165 A243764 A201821 * A122151 A294384 A216868

Adjacent sequences: A001053 A001054 A001055 * A001057 A001058 A001059

Keyword

nonn,easy,nice

Author

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

Status

approved