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A019489 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(3,7).

%I A019489 #28 Feb 16 2016 06:08:35
%S A019489 3,7,16,36,80,177,391,863,1904,4200,9264,20433,45067,99399,219232,
%T A019489 483532,1066464,2352161,5187855,11442175,25236512,55660880,122763936,
%U A019489 270764385,597189651,1317143239,2905050864,6407291380,14131726000,31168502865,68744297111
%N A019489 Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(3,7).
%C A019489 a(n) = A077852(n+1) (Barker's recurrence) is correct at least up to n=32000. - _R. J. Mathar_, Feb 11 2016
%C A019489 Not to be confused with the Pisot T(3,7) sequence, which is A020746. - _R. J. Mathar_, Feb 13 2016
%H A019489 Colin Barker, <a href="/A019489/b019489.txt">Table of n, a(n) for n = 0..1000</a>
%H A019489 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 A019489 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%F A019489 Empirical G.f.: -(x^3-x^2+2*x-3) / ((x-1)*(x^3+2*x-1)). [_Colin Barker_, Dec 21 2012]
%F A019489 a(n+1) = ceiling(a(n)^2/a(n-1))-1 for n>0. - _Bruno Berselli_, Feb 15 2016
%p A019489 A019489 := proc(n)
%p A019489         option remember;
%p A019489         if n <= 1 then
%p A019489                 op(n+1,[3,7]) ;
%p A019489         else
%p A019489                 a := procname(n-1)^2/procname(n-2) ;
%p A019489                 if type(a,'integer') then
%p A019489                         a-1 ;
%p A019489                 else
%p A019489                         floor(a) ;
%p A019489                 fi;
%p A019489         end if;
%p A019489 end proc: # _R. J. Mathar_, Feb 11 2016
%o A019489 (PARI) T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
%o A019489 T(3, 7, 30) \\ _Colin Barker_, Feb 16 2016
%K A019489 nonn
%O A019489 0,1
%A A019489 _R. K. Guy_