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A089983
1, 1, 1, 1, ... a, b, c, d, ab-cd, ...
2
1, 1, 1, 1, 0, 1, 1, -1, 1, 2, -3, 5, 17, -91, 1532, 139497, -213710951, 29812036392235, 6371158648631364574889, -189937213493701003981668660072118562, 1210120120447335073097142485947209203511752911347585124133
OFFSET
1,10
COMMENTS
Inspired by the formula for the determinant of a 2 X 2 matrix.
Sequence b(n,p) = a(n) (mod p), p prime, is a periodic sequence. Letting l(p) denotes the length of the period of b(n,p) we get l(2)=5, l(3)=11, l(5)=31... Is there any rule for l(p) ? - Benoit Cloitre, Nov 19 2003
LINKS
FORMULA
a(1)=a(2)=a(3)=a(4)=1, for n>4 a(n)=a(n-4)*a(n-3)-a(n-2)*a(n-1).
a(n) is asymptotic (in absolute value) to A^(phi^n) where phi=golden ratio and A=1.005384.. (follows same kind of behavior as A000301, A007660) - Benoit Cloitre, Nov 19 2003
MATHEMATICA
nxt[{a_, b_, c_, d_}]:={b, c, d, a b-c d}; NestList[nxt, {1, 1, 1, 1}, 20][[All, 1]] (* Harvey P. Dale, Oct 30 2021 *)
PROG
(PARI) a=b=c=d=1; for(n=5, 20, e=a*b-c*d; a=b; b=c; c=d; d=e; print1(e, ", "))
(Magma) I:=[1, 1, 1, 1]; [n le 4 select I[n] else -Self(n-1)*Self(n-2)+Self(n-3)*Self(n-4): n in [1..22]]; // Vincenzo Librandi, Mar 30 2014
CROSSREFS
Cf. A089984.
Sequence in context: A082979 A065952 A308316 * A370606 A072858 A276629
KEYWORD
sign,easy
AUTHOR
Ray Chandler, following a suggestion of Rainer Rosenthal, Nov 18 2003
STATUS
approved