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A174168 A (1,3) Somos-4 sequence.

%I A174168 #46 Oct 12 2024 21:31:18
%S A174168 1,2,5,17,109,706,9529,149057,3464585,141172802,5987285341,
%T A174168 540281496721,53055650250901,8453268527614082,2173247123169247025,
%U A174168 610193552206250342657,364280506115937098013649,225791960727416355391439746
%N A174168 A (1,3) Somos-4 sequence.
%C A174168 Hankel transform of A128720.
%H A174168 Reinhard Zumkeller, <a href="/A174168/b174168.txt">Table of n, a(n) for n = 0..100</a>
%H A174168 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry1/barry95r.html">Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences </a>, J. Int. Seq. 13 (2010) #10.7.2.
%H A174168 Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv:1107.5490 [math.CO], 2011.
%H A174168 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H A174168 P. Rajkovic, P. Barry, and N. Savic, <a href="http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-219-228.pdf">Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants</a>, Math. Balkanica, Vol. 26 (2012), Fasc. 1-2.
%F A174168 a(n) = (a(n-1)*a(n-3) + 3*a(n-2)^2)/a(n-4) [index corrected by _Franklin T. Adams-Watters_, Oct 26 2011].
%F A174168 a(n) = a(-2-n), a(n)*a(n-5) = -3*a(n-1)*a(n-4) + 16*a(n-2)*a(n-3) for all n in Z. - _Michael Somos_, Apr 30 2020
%p A174168 A174168 := proc(n)
%p A174168         if n <=4 then
%p A174168                 op(n+1,[1,2,5,17,109]) ;
%p A174168         else
%p A174168                 (procname(n-1)*procname(n-3)+3*procname(n-2)^2)/procname(n-4) ;
%p A174168         end if;
%p A174168 end proc: # _R. J. Mathar_, Oct 26 2011
%t A174168 RecurrenceTable[{a[0]==1,a[1]==2,a[2]==5,a[3]==17,a[n]==(a[n-1]a[n-3]+ 3a[n-2]^2)/a[n-4]},a[n],{n,20}] (* _Harvey P. Dale_, Oct 25 2011 *)
%t A174168 a[ n_] := Which[n < -1, a[-2 - n], n < 2, {1, 1, 2}[[n + 2]], True, (a[n - 1] a[n - 3] + 3 a[n - 2]^2)/a[n - 4]]; (* _Michael Somos_, Apr 30 2020 *)
%o A174168 (Haskell)
%o A174168 a174168 n = a174168_list !! n
%o A174168 a174168_list = [1,2,5,17] ++ zipWith div (zipWith (+)
%o A174168    (zipWith (*) (tail a174168_list) (drop 3 a174168_list))
%o A174168                 (map ((* 3) . (^ 2)) (drop 2 a174168_list))) a174168_list
%o A174168 -- _Reinhard Zumkeller_, Feb 13 2012
%o A174168 (PARI) m=20; v=concat([1,2,5,17], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] +3*v[n-2]^2)/v[n-4]); v \\ _G. C. Greubel_, Sep 25 2018
%o A174168 (Magma) I:=[1,2,5,17]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + 3*Self(n-2)^2)/Self(n-4): n in [1..20]]; // _G. C. Greubel_, Sep 25 2018
%Y A174168 Cf. A128720.
%K A174168 easy,nonn
%O A174168 0,2
%A A174168 _Paul Barry_, Mar 10 2010
%E A174168 More terms from _Sean A. Irvine_, Feb 12 2012