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A050512 a(n) = (a(n-1)*a(n-3) - a(n-2)^2) / a(n-4), with a(0) = 0, a(1) = a(2) = a(3) = 1, a(4) = -1.

%I A050512 #51 Jan 05 2025 19:51:36
%S A050512 0,1,1,1,-1,-2,-3,-1,7,11,20,-19,-87,-191,-197,1018,2681,8191,-5841,
%T A050512 -81289,-261080,-620551,3033521,14480129,69664119,-2664458,
%U A050512 -1612539083,-7758440129,-37029252553,181003520899,1721180313660,12437589708389,19206818781913
%N A050512 a(n) = (a(n-1)*a(n-3) - a(n-2)^2) / a(n-4), with a(0) = 0, a(1) = a(2) = a(3) = 1, a(4) = -1.
%C A050512 From _Paul Barry_, May 31 2010: (Start)
%C A050512 a(n+1) is (-1)^binomial(n,2) times the Hankel transform of the sequence with g.f.
%C A050512 1/(1-x/(1+x^2/(1-x^2/(1-2x^2/(1+(3/4)x^2/(1+(2/9)x^2/(1+21)x^2/(1-... where
%C A050512 -1,1,2,-3/4,-2/9,21,... are the x-coordinates of the multiples of z=(0,0) on the elliptic curve E: y^2 - 2xy - y = x^3-x. (End)
%C A050512 This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = -1.
%C A050512 The elliptic curve y^2 + y = x^3 + x^2 (LMFDB label 43.a1) has infinite order point P = (0, 0). The x coordinate of n*P has denominator a(n)^2. - _Michael Somos_, Feb 14 2023
%H A050512 Reinhard Zumkeller, <a href="/A050512/b050512.txt">Table of n, a(n) for n = 0..250</a>
%H A050512 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A050512 Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.
%H A050512 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H A050512 LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/43/a/1">Elliptic Curve 43.a1 (Cremona label 43a1)</a>
%F A050512 a(2*n + 1) = a(n + 2) * a(n)^3 - a(n - 1) * a(n + 1)^3 for all n in Z.
%F A050512 a(2*n) = a(n + 2) * a(n) * a(n - 1)^2 - a(n) * a(n - 2) * a(n + 1)^2 for all n in Z.
%F A050512 0 = a(n)*a(n+5) - a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z. - _Michael Somos_, Jul 07 2014
%F A050512 0 = a(n)*a(n+6) + a(n+1)*a(n+5) - 2*a(n+2)*a(n+4) for all n in Z. - _Michael Somos_, Jul 07 2014
%F A050512 a(n) = -a(-n) for all n in Z. - _Michael Somos_, Feb 14 2023
%e A050512 G.f. = x + x^2 + x^3 - x^4 - 2*x^5 - 3*x^6 - x^7 + 7*x^8 + 11*x^9 + 20*x^10 + ...
%t A050512 a[n_?OddQ] := a[n] = a[(n-1)/2]^3*a[(n+3)/2] - a[(n-3)/2]*a[(n+1)/2]^3; a[n_?EvenQ] := a[n] = (a[n/2-1]^2*a[n/2+2] - a[n/2-2]*a[n/2+1]^2)*a[n/2]; a[0] = 0; a[1] = a[2] = a[3] = 1; a[4] = -1; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 29 2011 *)
%t A050512 Join[{0},RecurrenceTable[{a[1]==a[2]==a[3]==1,a[4]==-1,a[n]==(a[n-1] a[n-3]-a[n-2]^2)/a[n-4]},a,{n,30}]] (* _Harvey P. Dale_, Mar 23 2012 *)
%o A050512 (PARI) an=vector(200); for(n=1,4,an[ n ]=[ 1,1,1,-1 ][ n ]); for(n=5, length(an),an[ n ]=(an[ n-1 ]*an[ n-3 ]-an[ n-2 ]^2)/an[ n-4 ]); a(n) =sign(n)*an[ abs(n)+(n==0) ]
%o A050512 (PARI) {a(n) = my(an); if( n<0, -a(-n), if( n==0, 0, an = vector( max(4, n), i, 1); an[4] = -1; for( k=5, n, an[k] = (an[k-1] * an[k-3] - an[k-2]^2) / an[k-4]); an[n]))}; /* _Michael Somos_, Jul 07 2014 */
%o A050512 (Haskell)
%o A050512 a050512 n = a050512_list !! n
%o A050512 a050512_list = 0 : 1 : 1 : 1 : (-1) : zipWith div (zipWith (-) (zipWith (*)
%o A050512    (drop 4 a050512_list) (drop 2 a050512_list))
%o A050512      (map (^ 2) (drop 3 a050512_list))) (tail a050512_list)
%o A050512 -- _Reinhard Zumkeller_, Nov 02 2011
%Y A050512 Cf. A006769.
%K A050512 sign,easy,nice
%O A050512 0,6
%A A050512 _Michael Somos_, Dec 28 1999