cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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.

Original entry on oeis.org

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

Views

Author

Michael Somos, Feb 04 2023

Keywords

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

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)

A241594 a(n) = -(8*a(n-4)*a(n-1)+57*a(n-3)*a(n-2))/a(n-5) with initial values 1, 0, -1, 1, 8, 57, -455.

Original entry on oeis.org

1, 0, -1, 1, 8, 57, -455, 22352, 47767, 69739671, 3385862936, 1747973613295, -632038062613231, 319807929289790304, 778756000716629557903, 186509371006506937278833, -7581885296067966478838810840, -17592286469464275110206466526327, 594744699237794019378328459208828297
Offset: 0

Views

Author

N. J. A. Sloane, May 18 2014

Keywords

Links

Crossrefs

See A241595 for another version.
Cf. A360381.

Programs

  • Magma
    I:=[22352, 47767, 69739671, 3385862936, 1747973613295]; [1, 0, -1, 1, 8, 57, -455] cat [n le 5 select I[n] else -(8*Self(n-4)*Self(n-1) + 57*Self(n-3)*Self(n-2))/Self(n-5): n in [1..20]]; // G. C. Greubel, Aug 08 2018
  • Maple
    f:=proc(n) option remember;
if n <= 2 then 1-n
elif n=3 then 1
elif n=4 then 8
elif n=5 then 57
elif n=6 then -455
else -(8*f(n-4)*f(n-1)+57*f(n-3)*f(n-2))/f(n-5); fi; end;
[seq(f(n),n=0..30)];
  • Mathematica
    a[n_] := a[n] = If[n <= 6, {1, 0, -1, 1, 8, 57, -455}[[n + 1]], -(8*a[n - 4]*a[n - 1] + 57*a[n - 3]*a[n - 2])/a[n - 5]];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Dec 02 2017 *)
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,-(8b e+57c d)/a}; Join[{1,0},NestList[nxt,{-1,1,8,57,-455},20] [[;;,1]]] (* Harvey P. Dale, Aug 14 2023 *)
  • PARI
    {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, n), 'x, 1) * (-1)^(n%4!=2)}; /* Michael Somos, Mar 01 2025 */
  • PARI
    {a(n) = my(E = ellinit([1, 1, 0, -2, 0])); n--; subst(elldivpol(E, 2*n), 'x, 2) * (-1)^(n + (n%4==2))/6^(n^2 + 1)}; /* Michael Somos, Mar 02 2025 */

Formula

-a(n+1) = A360381(2*n)*(-1)^(n%4!=0). a(n+2)*a(n-2) = -(-1)^n*a(n+1)*a(n-1) + 8*a(n)^2 for all n in Z. - Michael Somos, Mar 01 2025
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.