A064183 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = q(n) and A064526(n) = p(n).

1, 1, 1, 2, 3, 10, 39, 490, 20631, 10349290, 213941840151, 2214253254659846890, 473721461633379426414550183191, 1048939288228833100615882755549676600679754298090

Offset

0,4

Links

Table of n, a(n) for n=0..13.

Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540.

Formula

a(n) = (a(n-1) + a(n-2))*a(n-2) for n >= 2.

Lim_{n -> infinity} a(n)/a(n-1)^phi = 1, where phi = A001622. - Gerald McGarvey, Aug 29 2004

a(n) ~ c^(phi^n), where c = 1.23642417842410860616065684299168229758826316461949675490684055924721259... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

Mathematica

Flatten[{1, RecurrenceTable[{a[n]==(a[n-1]+a[n-2])*a[n-2], a[1]==1, a[2]==1}, a, {n, 1, 10}]}] (* Vaclav Kotesovec, May 21 2015 *)

Prog

(PARI) {a(n) = local(v); if( n<3, n>=0, v = [1, 1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[2])}
(PARI) {a(n) = if( n<3, n>=0, (a(n-1) + a(n-2)) * a(n-2))}

Crossrefs

Cf. A001622, A003686, A064526, A064847, A070231, A070233, A070234, A094303, A236394.

Sequence in context: A164933 A003048 A008980 * A050381 A129716 A032293

Adjacent sequences: A064180 A064181 A064182 * A064184 A064185 A064186

Keyword

nonn,easy

Author

Michael Somos, Sep 20 2001

Status

approved