A210098 Somos-5 sequence variant: a(n) = (a(n-1) * a(n-4) - a(n-2) * a(n-3)) / a(n-5), a(0) = 0, a(1) = a(2) = a(3) = a(4) = 1, a(5) = 2.

0, 1, 1, 1, 1, 2, 1, -1, -3, -5, -4, -11, -13, -7, 23, 86, 87, 199, 415, 799, -152, -4159, -8063, -17047, -38727, -155366, -142471, 445015, 2309453, 7627979, 13609844, 81138437, 187790979, 142104721, -1743980081, -12357952274, -25547499185, -134098256401

Offset

0,6

Comments

This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

K. S. Brown, A Quasi-Periodic Sequence

Index to divisibility sequences

Formula

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

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

Example

G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 - x^7 - 3*x^8 - 5*x^9 - 4*x^10 + ...

Maple

a:= proc(n) a(n):= `if`(n<6, [0, 1$4, 2][n+1],
      (a(n-1)*a(n-4) -a(n-2)*a(n-3)) / a(n-5))
    end:
seq (a(n), n=0..40);  # Alois P. Heinz, Oct 20 2012

Mathematica

Join[{0}, RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1, a[5]==2, a[n] == (a[n-1]a[n-4]-a[n-2]a[n-3])/a[n-5]}, a, {n, 40}]] (* Harvey P. Dale, Oct 20 2012 *)

Prog

(PARI) {a(n) = my(v, m); if( n==0, 0, m = abs(n); sign(n) * if( m<6, 1 + (m>4), v = vector( m, i, 1); v[5] = 2; for( i=6, m, v[i] = (v[i-1] * v[i-4] - v[i-2] * v[i-3]) / v[i-5]); v[m]))};
(Magma) I:=[1, 1, 1, 1, 2]; [n le 5 select I[n] else (Self(n-1)*Self(n-4) - Self(n-2)*Self(n-3))/Self(n-5): n in [1..50]]; // G. C. Greubel, Aug 11 2018

Crossrefs

Cf. A006721.

Sequence in context: A187066 A187065 A174620 * A241188 A145236 A162206

Adjacent sequences: A210095 A210096 A210097 * A210099 A210100 A210101

Keyword

sign

Author

Michael Somos, Mar 17 2012

Status

approved