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A061021 a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.

%I A061021 #40 Sep 13 2023 11:14:41
%S A061021 3,3,3,6,15,87,1299,112998,146784315,16586334025071,
%T A061021 2434613678231239448367,40381315689150066251526220641224742,
%U A061021 98312903521778500654864668915856114278134197773017871243
%N A061021 a(n) = a(n-1)*a(n-2) - a(n-3) with a(0) = a(1) = a(2) = 3.
%C A061021 Any three consecutive terms are a solution to the Diophantine equation x^2 + y^2 + z^2 = xyz.
%H A061021 Harry J. Smith, <a href="/A061021/b061021.txt">Table of n, a(n) for n = 0..17</a>
%H A061021 Loren C. Larson, <a href="https://www.jstor.org/stable/1559041">Solution to Problem Proposal 701</a>, College Mathematics Journal 33 (2002), pp. 241-242.
%H A061021 Edward T. H. Wang, <a href="http://www.jstor.org/stable/2687476">Problem Proposal 701</a>, College Mathematics Journal 32 (2001), p. 211.
%F A061021 From _Jon E. Schoenfield_, May 12 2019: (Start)
%F A061021 It appears that, for n >= 1,
%F A061021   a(n) = ceiling(e^(c0*phi^n - c1/(-phi)^n))
%F A061021 where
%F A061021   phi = (1 + sqrt(5))/2,
%F A061021    c0 = 0.4004033011137849744572073756789830081726425559860...
%F A061021    c1 = 0.2798639753144007577581523025628820390768226527315...
%F A061021 (End)
%t A061021 RecurrenceTable[{a[n] == a[n - 1] a[n - 2] - a[n - 3], a[0] == a[1] == a[2] == 3}, a, {n, 0, 12}] (* _Michael De Vlieger_, Aug 21 2016 *)
%o A061021 (PARI) for (n=0, 17, if (n>2, a=a1*a2 - a3; a3=a2; a2=a1; a1=a, if (n==0, a=a3=3, if (n==1, a=a2=3, a=a1=3))); write("b061021.txt", n, " ", a)) \\ _Harry J. Smith_, Jul 16 2009
%o A061021 (Haskell)
%o A061021 a061021 n = a061021_list !! n
%o A061021 a061021_list = 3 : 3 : 3 : zipWith (-)
%o A061021 (tail $ zipWith (*) (tail a061021_list) a061021_list) a061021_list
%o A061021 -- _Reinhard Zumkeller_, Mar 25 2015
%Y A061021 Cf. A022405, A061292, A072878, A072879, A072880, A074394, A178768.
%K A061021 easy,nonn
%O A061021 0,1
%A A061021 _Stephen G Penrice_, May 23 2001
%E A061021 More terms from _Erich Friedman_, Jun 03 2001
%E A061021 Name clarified by _Petros Hadjicostas_, May 11 2019