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A022024 Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(6,66).

%I A022024 #40 Feb 14 2024 10:46:49
%S A022024 6,66,727,8009,88232,972018,10708349,117969769,1299627646,14317498734,
%T A022024 157730385799,1737655093709,19143078927992,210891949829430,
%U A022024 2323315631208341,25595076182769253,281971126093205254,3106367622527151978,34221659288246953735,377006879658404795777
%N A022024 Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(6,66).
%C A022024 This coincides with the linearly recurrent sequence defined by the expansion of (6 - 5*x^2)/(1 - 11*x - x^2 + 9*x^3) only up to n <= 169. - _Bruno Berselli_, Feb 11 2016
%H A022024 Colin Barker, <a href="/A022024/b022024.txt">Table of n, a(n) for n = 0..950</a>
%F A022024 a(n+1) = floor(a(n)^2/a(n-1))+1 for all n > 0. - _M. F. Hasler_, Feb 10 2016
%p A022024 A022024 := proc(n)
%p A022024     option remember;
%p A022024     if n <= 1 then
%p A022024         op(n+1,[6,66]) ;
%p A022024     else
%p A022024         a := procname(n-1)^2/procname(n-2) ;
%p A022024         if type(a,'integer') then
%p A022024             a+1 ;
%p A022024         else
%p A022024             ceil(a) ;
%p A022024         fi;
%p A022024     end if;
%p A022024 end proc: # _R. J. Mathar_, Feb 10 2016
%t A022024 a[n_] := a[n] = Switch[n, 0, 6, 1, 66, _, Floor[a[n-1]^2/a[n-2]]+1];
%t A022024 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 08 2024 *)
%o A022024 (PARI) a=[6,66];for(n=2,30,a=concat(a,a[n]^2\a[n-1]+1));a \\ _M. F. Hasler_, Feb 10 2016
%o A022024 (Python)
%o A022024 def a(n):
%o A022024     if n == 0: return 6
%o A022024     prev_1, prev_2 = 66, 6
%o A022024     for i in range(2, n + 1):
%o A022024         prev_2, prev_1 = prev_1, (prev_1 ** 2) // prev_2 + 1
%o A022024     return prev_1 # _Paul Muljadi_, Feb 12 2024
%Y A022024 Cf. A022018 - A022025, A022026 - A022032.
%K A022024 nonn
%O A022024 0,1
%A A022024 _R. K. Guy_
%E A022024 Double-checked and extended to 3 lines of data by _M. F. Hasler_, Feb 10 2016