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A360381 Generalized Somos-5 sequence a(n) = (a(n-1)*a(n-4) + a(n-2)*a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.

%I A360381 #33 Aug 20 2025 00:29:53
%S A360381 0,1,-1,1,1,-7,8,-1,-57,391,-455,-2729,22352,-175111,47767,8888873,
%T A360381 -69739671,565353361,3385862936,-195345149609,1747973613295,
%U A360381 -4686154246801,-632038062613231,34045765616463119,-319807929289790304,-11453004955077020783
%N A360381 Generalized Somos-5 sequence a(n) = (a(n-1)*a(n-4) + a(n-2)*a(n-3))/a(n-5) = -a(-n), a(1) = 1, a(2) = -1, a(3) = a(4) = 1, a(5) = -7.
%C A360381 This has the same recurrence as Somos-5 (A006721) with different initial values.
%C A360381 The elliptic curve y^2 + xy = x^3 + x^2 - 2x (LMFDB label 102.a1) has infinite order point P = (2, 2). The x and y coordinates of n*P have denominators a(n)^2 and |a(n)^3| respectively.
%C A360381 If b(2*n) = 6^(1/4)*a(2*n), b(2*n+1) = a(2*n+1), then b(n) is a generalized Somos-4 sequence with b(n+2)*b(n-2) = 6^(1/2)*b(n+1)*b(n-1) - b(n)*b(n) for all n in Z.
%C A360381 This is the sequence T_n in the Hone 2022 paper.
%H A360381 Andrew N. W. Hone, <a href="https://doi.org/10.1007/s40879-022-00586-w">Heron triangles with two rational medians and Somos-5 sequences</a>, European Journal of Mathematics, 8 (2022), 1424-1486; arXiv:<a href="https://arxiv.org/abs/2107.03197">2107.03197</a> [math.NT], 2021-2022.
%H A360381 Andrew N. W. Hone, <a href="https://arxiv.org/abs/2401.05581">Heron triangles and the hunt for unicorns</a>, arXiv:2401.05581 [math.NT], 2024.
%H A360381 LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/102/a/1">Elliptic Curve 102.a1 (Cremona label 102a1)</a>
%F A360381 a(2*n) = -A241595(n+1), a(n) = -a(-n) for all n in Z.
%F A360381 From _Michael Somos_, Aug 19 2025: (Start)
%F A360381 Let S(n) = A006721(n+2) as in Hone. We have for all n in Z:
%F A360381 S(2*n) = S(n-1)*S(n)*a(n-1)*a(n+2) - S(n-2)*S(n+1)*a(n)*a(n+1).
%F A360381 S(2*n+1) = S(n)*S(n+1)*a(n-1)*a(n+2) - S(n-1)*S(n+2)*a(n)*a(n+1).
%F A360381 a(2*n) = a(n)*(a(n-2)*a(n+1)^2 - a(n+2)*a(n-1)^2).
%F A360381 a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n+2)*a(n+1)^2*a(n-2).
%F A360381 S(n-3)*S(n) = S(n-2)*S(n-1) - a(n-2)*a(n-1).
%F A360381 a(n-3)*a(n) = S(n-2)*S(n-1) + a(n-2)*a(n-1).
%F A360381 (End)
%e A360381 5*P = (50/49, 20/343) and a(5) = -7, 6*P = (121/64, -1881/512) and a(6) = 8.
%t A360381 a[0] = 0; a[1] = a[3] = a[4] = 1; a[2] = -1; a[5] = -7;
%t A360381 a[n_?Negative] := -a[-n];
%t A360381 a[n_] := a[n] = (a[n-1] a[n-4] + a[n-2] a[n-3]) / a[n-5]; (* _Andrey Zabolotskiy_, Feb 05 2023 *)
%t A360381 a[ n_] := Module[{A = Table[1, Max[5, Abs[n]]]}, A[[2]] = -1; A[[5]] = -7; Do[ A[[k]] = (A[[k-1]]*A[[k-4]] + A[[k-2]]*A[[k-3]])/A[[k-5]], {k, 6, Length[A]}]; If[n==0, 0, Sign[n]*A[[Abs[n]]] ]];
%o A360381 (PARI) {a(n) = my(A = vector(max(5, abs(n)), k, 1)); A[2] = -1; A[5] = -7; for(k=6, #A, A[k] = (A[k-1]*A[k-4] + A[k-2]*A[k-3])/A[k-5]); if(n==0, 0, sign(n)*A[abs(n)])};
%o A360381 (PARI) {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); subst(elldivpol(E, n), 'x, 2) *(-1)^(n-1) / 6^((n-1)%2 + n^2\4)}; /* _Michael Somos_, Mar 01 2025 */
%Y A360381 Cf. A006721, A241595.
%K A360381 sign,changed
%O A360381 0,6
%A A360381 _Michael Somos_, Feb 04 2023