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A018908 Define sequence S(a_0,a_1) by a_{n+2} is least integer such that a_{n+2}/a_{n+1}>a_{n+1}/a_n for n >= 0. This is S(3,4).

%I A018908 #18 Jul 13 2023 09:26:09
%S A018908 3,4,6,10,17,29,50,87,152,266,466,817,1433,2514,4411,7740,13582,23834,
%T A018908 41825,73397,128802,226031,396656,696082,1221538,2143649,3761841,
%U A018908 6601570,11584947,20330164,35676950,62608682,109870577,192809421,338356946,593775047
%N A018908 Define sequence S(a_0,a_1) by a_{n+2} is least integer such that a_{n+2}/a_{n+1}>a_{n+1}/a_n for n >= 0. This is S(3,4).
%H A018908 Alois P. Heinz, <a href="/A018908/b018908.txt">Table of n, a(n) for n = 0..1000</a>
%H A018908 D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
%H A018908 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%p A018908 a:= proc(n) option remember; `if`(n<2, [3, 4][n+1],
%p A018908        1 +floor(a(n-1)^2/a(n-2)))
%p A018908     end:
%p A018908 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 05 2014
%t A018908 a[n_] := a[n] = Switch[n, 0, 3, 1, 4, _, 1 + Floor[a[n-1]^2/a[n-2]]];
%t A018908 a /@ Range[0, 50] (* _Jean-François Alcover_, Nov 16 2020, after _Alois P. Heinz_ *)
%K A018908 nonn
%O A018908 0,1
%A A018908 _R. K. Guy_