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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(3,6).
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%I #16 Jul 13 2023 09:26:39

%S 3,6,13,29,65,146,328,737,1657,3726,8379,18843,42375,95295,214305,

%T 481942,1083821,2437364,5481296,12326680,27721007,62340730,140195723,

%U 315280889,709023335,1594495915,3585801902,8063975053,18134770251,40782602860,91714461944

%N 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(3,6).

%H Alois P. Heinz, <a href="/A018909/b018909.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>

%p a:= proc(n) option remember; `if`(n<2, [3, 6][n+1],

%p 1 +floor(a(n-1)^2/a(n-2)))

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, May 05 2014

%t a[n_] := a[n] = Switch[n, 0, 3, 1, 6, _, 1 + Floor[a[n-1]^2/a[n-2]]];

%t a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 16 2020, after _Alois P. Heinz_ *)

%K nonn

%O 0,1

%A _R. K. Guy_