A064183 - OEIS

%I #33 Mar 12 2021 22:24:42

%S 1,1,1,2,3,10,39,490,20631,10349290,213941840151,2214253254659846890,

%T 473721461633379426414550183191,

%U 1048939288228833100615882755549676600679754298090

%N 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).

%H Michael Somos and R. Haas, <a href="http://www.jstor.org/stable/3647911">A linked pair of sequences implies the primes are infinite</a>, Amer. Math. Monthly, 110(6) (2003), 539-540.

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

%F Lim_{n -> infinity} a(n)/a(n-1)^phi = 1, where phi = A001622. - _Gerald McGarvey_, Aug 29 2004

%F 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

%t 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 *)

%o (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])}

%o (PARI) {a(n) = if( n<3, n>=0, (a(n-1) + a(n-2)) * a(n-2))}

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

%K nonn,easy

%O 0,4

%A _Michael Somos_, Sep 20 2001