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A019480 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(4,12) (agrees with A019481 for n <= 19 only).

%I A019480 #25 Jul 13 2023 09:37:51
%S A019480 4,12,37,115,358,1115,3473,10818,33697,104963,326950,1018419,3172281,
%T A019480 9881362,30779529,95875387,298642966,930245227,2897627873,9025842914,
%U A019480 28114666162,87574585658,272786737320,849705465331,2646753962113,8244393877392,25680524664755
%N A019480 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(4,12) (agrees with A019481 for n <= 19 only).
%H A019480 Alois P. Heinz, <a href="/A019480/b019480.txt">Table of n, a(n) for n = 0..2025</a>
%H A019480 D. W. Boyd, <a href="http://www.researchgate.net/publication/258834801">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 A019480 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%p A019480 a:= proc(n) option remember;
%p A019480       `if`(n<2, [4, 12][n+1], floor(a(n-1)^2/a(n-2))+1)
%p A019480     end:
%p A019480 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 18 2015
%t A019480 S[a_, b_, n_] := Block[{s = {a, b}, k}, Do[k = Last@ s + 1; While[k/s[[i - 1]] <= s[[i - 1]]/s[[i - 2]], k++]; AppendTo[s, k], {i, 3, n}]; s]; S[4, 12, 14] (* _Michael De Vlieger_, Feb 15 2016 *)
%o A019480 (PARI) S(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=a[n-1]^2\a[n-2]+1); a
%o A019480 S(4, 12, 40) \\ _Colin Barker_, Feb 15 2016
%K A019480 nonn
%O A019480 0,1
%A A019480 _R. K. Guy_