|
%I #24 Feb 05 2023 13:50:23
%S 0,1,-1,1,1,-7,8,-1,-57,391,-455,-2729,22352,-175111,47767,8888873,
%T -69739671,565353361,3385862936,-195345149609,1747973613295,
%U -4686154246801,-632038062613231,34045765616463119,-319807929289790304,-11453004955077020783
%N 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 This has the same recurrence as Somos-5 (A006721) with different initial values.
%C 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 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 This is the sequence T_n in the Hone 2022 paper.
%H A. 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 LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/102/a/1">Elliptic Curve 102.a1 (Cremona label 102a1)</a>
%F a(2*n) = -A241595(n+1), a(n) = -a(-n) for all n in Z.
%e 5*P = (50/49, 20/343) and a(5) = -7, 6*P = (121/64, -1881/512) and a(6) = 8.
%t a[0] = 0; a[1] = a[3] = a[4] = 1; a[2] = -1; a[5] = -7;
%t a[n_?Negative] := -a[-n];
%t 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 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 (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)])};
%Y Cf. A006721, A241595.
%K sign
%O 0,6
%A _Michael Somos_, Feb 04 2023
|
