A089984 1, 1, 1, 1, ... a, b, c, d, ac-bd, ...

1, 1, 1, 1, 0, 1, -1, 1, -1, 0, 1, 1, -1, 1, -2, -1, 3, 5, -1, -2, 7, -3, -13, 97, 200, 2309, -226573, 45538573, -105193879657, -23833987746960404, 1085365814730154781188953, 114173840897460294190477827374165629

Offset: 1

Author

Ray Chandler, following a suggestion of Rainer Rosenthal, Nov 18 2003

Keywords

• sign
• easy

Comments

Inspired by the formula for the determinant of a 2 X 2 matrix.

Sequence b(n,p) = a(n) (mod p), p prime, n>4, is a periodic sequence. Letting l(p) denotes the length of the period of b(n,p) is there any rule for l(p) ? - Benoit Cloitre, Nov 19 2003

Crossrefs

Cf. A089983.

Programs

Mathematica

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==a[n-4]a[n-2]- a[n-3] a[n-1]}, a,{n,40}] (* Harvey P. Dale, Aug 24 2019 *)

PARI

a=b=c=d=1;for(n=5,30,e=b*d-a*c;a=b;b=c;c=d;d=e;print1(e,","))

Formula

a(1)=a(2)=a(3)=a(4)=1, for n>4 a(n)=a(n-4)*a(n-2)-a(n-3)*a(n-1).

a(n) is asymptotic (in absolute value) to B^(r^n) where r is the real root of 1+x^2-x^3 and B>1. - Benoit Cloitre, Nov 19 2003

Extensions

Corrected by Harvey P. Dale, Aug 24 2019