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.

0, 1, -1, 1, 1, -7, 8, -1, -57, 391, -455, -2729, 22352, -175111, 47767, 8888873, -69739671, 565353361, 3385862936, -195345149609, 1747973613295, -4686154246801, -632038062613231, 34045765616463119, -319807929289790304, -11453004955077020783

Offset: 0

Author

Michael Somos, Feb 04 2023

Keywords

• sign
• changed

Comments

This has the same recurrence as Somos-5 (A006721) with different initial values.

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.

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.

This is the sequence T_n in the Hone 2022 paper.

Examples

			5*P = (50/49, 20/343) and a(5) = -7, 6*P = (121/64, -1881/512) and a(6) = 8.
		

Links

• Andrew N. W. Hone, Heron triangles with two rational medians and Somos-5 sequences, European Journal of Mathematics, 8 (2022), 1424-1486; arXiv:2107.03197 [math.NT], 2021-2022.
• Andrew N. W. Hone, Heron triangles and the hunt for unicorns, arXiv:2401.05581 [math.NT], 2024.
• LMFDB, Elliptic Curve 102.a1 (Cremona label 102a1)

Crossrefs

Cf. A006721, A241595.

Programs

Mathematica

a[0] = 0; a[1] = a[3] = a[4] = 1; a[2] = -1; a[5] = -7;
a[n_?Negative] := -a[-n];
a[n_] := a[n] = (a[n-1] a[n-4] + a[n-2] a[n-3]) / a[n-5]; (* Andrey Zabolotskiy, Feb 05 2023 *)
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]]] ]];

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)])};

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 */

Formula

a(2*n) = -A241595(n+1), a(n) = -a(-n) for all n in Z.

From Michael Somos, Aug 19 2025: (Start)

Let S(n) = A006721(n+2) as in Hone. We have for all n in Z:

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).

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).

a(2*n) = a(n)*(a(n-2)*a(n+1)^2 - a(n+2)*a(n-1)^2).

a(2*n+1) = a(n-1)*a(n)^2*a(n+3) - a(n+2)*a(n+1)^2*a(n-2).

S(n-3)*S(n) = S(n-2)*S(n-1) - a(n-2)*a(n-1).

a(n-3)*a(n) = S(n-2)*S(n-1) + a(n-2)*a(n-1).

(End)