A074394 a(n) = a(n-1)*a(n-2) - a(n-3) with a(1) = 1, a(2) = 2, and a(3) = 3.

1, 2, 3, 5, 13, 62, 801, 49649, 39768787, 1974480504962, 78522694637486171445, 155041529758800625329015665441303, 12174278697379026530632791354719900462885271361687873

Offset

1,2

Comments

All consecutive quadruples are pairwise coprime. Multiples of 2 occur when n=2 mod 4, multiples of 3 when n=3 mod 4, multiples of 5 when n=4 mod 7, multiples of 7 when n=10 mod 14, multiples of 9 when n=7 or 11 mod 24, multiples of 10 when n=18 mod 28. Multiples of 4, 6 and 8 never occur.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..19

Formula

Lim_{n->infinity} a(n+1)/a(n)^phi = 1, where phi is the golden ratio (1+sqrt(5))/2 = A001622. - Benoit Cloitre, Sep 26 2002

From Jon E. Schoenfield, May 13 2019: (Start)

It appears that, for n >= 2,

a(n) = ceiling(e^(c*phi^n - d/(-phi)^n))

where

phi = (1 + sqrt(5))/2

c = 0.230193077518834725477008740044380256486365499661...

d = 0.067704372842879037264190305626317036100889750046...

(End)

Example

a(6) = a(5)*a(4) - a(3) = 13*5 - 3 = 62.

Mathematica

nxt[{a_, b_, c_}]:={b, c, b*c-a}; NestList[nxt, {1, 2, 3}, 15][[All, 1]] (* Harvey P. Dale, Jan 21 2023 *)

Prog

(Haskell)
a074394 n = a074394_list !! (n-1)
a074394_list = 1 : 2 : 3 : zipWith (-)
   (tail $ zipWith (*) (tail a074394_list) a074394_list) a074394_list
-- Reinhard Zumkeller, Mar 25 2015

Crossrefs

Cf. A001622, A022405, A061021, A072878, A072879, A072880.

Sequence in context: A103594 A042695 A288943 * A056162 A265785 A326372

Adjacent sequences: A074391 A074392 A074393 * A074395 A074396 A074397

Keyword

nonn

Author

Henry Bottomley, Sep 24 2002

Status

approved