A210098 -id:A210098

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A277279

Somos-4 sequence variant: a(n) = (a(n-1) * a(n-3) - a(n-2)^2) / a(n-4), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = -1.

+10
2

1, 1, 2, -1, -5, -11, -7, 86, 199, 799, -4159, -17047, -155366, 445015, 7627979, 81138437, 142104721, -12357952274, -134098256401, -2117060496481, 57564521075233, 987319483194481, 40297982292465650, -635283824578537969, -39106648195100243333

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..174

FORMULA

a(n) = A210098(2*n + 1).

0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 for all n in Z.

0 = a(n)*a(n+5) - a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z.

0 = a(n+1)^2*a(n+2)^2 - a(n)^2*a(n+3)^2 - a(n)*a(n+2)^3 - a(n+1)^3*a(n+3) - 2*a(n)*a(n+1)*a(n+2)*a(n+3) for all n in Z.

a(n) = -a(-1-n) for all n in Z. - Michael Somos, Mar 14 2020

EXAMPLE

G.f. = 1 + x + 2*x^2 - x^3 - 5*x^4 - 11*x^5 - 7*x^6 + 86*x^7 + 199*x^8 + ...

MATHEMATICA

a[ n_] := a[n] = Which[ n < 0, -a[-1 - n], n < 3, 1 + Boole[n > 1], True, (a[n - 1] a[n - 3] - a[n - 2]^2) / a[n - 4]];

RecurrenceTable[{a[0]==a[1]==1, a[2]==2, a[3]==-1, a[n]==(a[n-1]a[n-3]-a[n-2]^2)/ a[n-4]}, a, {n, 30}] (* Harvey P. Dale, Nov 28 2019 *)

PROG

(PARI) {a(n) = my(v, m); if( n<0, -a(-1 -n), n<3, 1 + (n>1), v = vector( m=n+2, i, (-1)^(i<3) + (i==5)); for( i=6, m, v[i] = (v[i-1] * v[i-3] - v[i-2]^2) / v[i-4]); v[m])};

CROSSREFS

Cf. A006720, A210098.

KEYWORD

sign

AUTHOR

Michael Somos, Oct 08 2016

STATUS

approved

A377310

Divisibility sequence associated with elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 and point (1, 0).

+10
0

0, 1, 1, 1, -3, -4, -13, 23, 87, 415, -152, -8063, -38727, -142471, 2309453, 2309453, 13609844, 187790979, -1743980081, -25547499185, -575984295329, 1873521429456, 217675476797921, 5045023692031697, 65853623974941521

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

A bisection of A210098. The other bisection is A277279.

The elliptic curve y^2 + y = x^3 - x^2 - 2x + 2 has LMFDB label 57.a1 (Cremona label 57a1).

This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 1, y = 1, z = -3.

LINKS

Table of n, a(n) for n=0..24.

C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.

LMFDB, Elliptic Curve 57a1

FORMULA

a(n) = A210098(2*n).

a(n) = -a(-n) for all n in Z.

0 = a(n)*a(n+4) - a(n+1)*a(n+3) + a(n+2)^2 for all n in Z.

0 = a(n)*a(n+5) - a(n+1)*a(n+4) - 3*a(n+2)*a(n+3) for all n in Z.

0 = a(n+1)^2*a(n+2)^2 - a(n)^2*a(n+3)^2 - a(n)*a(n+2)^3 - a(n+1)^3*a(n+3) - 2*a(n)*a(n+1)*a(n+2)*a(n+3) for all n in Z.

EXAMPLE

G.f. = x + x^2 + x^3 - 3*x^4 - 4*x^5 - 13*x^6 + 23*x^7 + 87*x^8 + 415*x^9 + ...

MATHEMATICA

a[ n_] := a[n] = Which[ n<0, -a[-n], n<5, {0, 1, 1, 1, -3}[[n+1]], True, (a[n-1]*a[n-3] - a[n-2]^2)/a[n-4]];

PROG

(PARI) {a(n) = my(v); if(n<0, -a(-n), n<5, [0, 1, 1, 1, -3][n+1], v = vector(n, i, if(i<5, a(i))); for(i=5, n, v[i] = (v[i-1]*v[i-3] - v[i-2]^2)/v[i-4]); v[n])};

(PARI) {a(n) = my(E = ellinit([0, -1, 1, -2, 2]), z); z = ellpointtoz(E, [1, 0]); -(-1)^n*round(ellsigma(E, n*z)/ellsigma(E, z)^(n^2))};

(PARI) {a(n) = my(E = ellinit([0, -1, 1, -2, 2])); sign(n) * subst( elldivpol( E, abs(n)), x, 1)};

CROSSREFS

Cf. A210098, A277279.

KEYWORD

sign,new

AUTHOR

Michael Somos, Oct 23 2024

STATUS

approved

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