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A018921 Define the generalized Pisot 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). This is T(4,8).

%I A018921 #40 Jul 13 2023 09:30:44
%S A018921 4,8,15,28,52,96,177,326,600,1104,2031,3736,6872,12640,23249,42762,
%T A018921 78652,144664,266079,489396,900140,1655616,3045153,5600910,10301680,
%U A018921 18947744,34850335,64099760,117897840,216847936,398845537,733591314,1349284788,2481721640
%N A018921 Define the generalized Pisot 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). This is T(4,8).
%C A018921 Not to be confused with the Pisot T(4,8) sequence, which is A020707. - _R. J. Mathar_, Feb 13 2016
%H A018921 Colin Barker, <a href="/A018921/b018921.txt">Table of n, a(n) for n = 0..1000</a>
%H A018921 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 A018921 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,-1).
%H A018921 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%F A018921 a(n) = 2*a(n-1) - a(n-4).
%F A018921 G.f.: (4-x^2-2*x^3) / ((1-x)*(1-x-x^2-x^3)). - _Colin Barker_, Feb 08 2012
%F A018921 a(n) = A008937(n+3) = A027084(n+3)+1. [first index correct by _R. J. Mathar_, Jun 24 2020]
%F A018921 a(n) = 2*a(n-1) - A008937(n). - _Vincenzo Librandi_, Feb 12 2016
%t A018921 RecurrenceTable[{a[1] == 4, a[2] == 8, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 40}] (* _Bruno Berselli_, Feb 17 2016 *)
%t A018921 LinearRecurrence[{2,0,0,-1},{4,8,15,28},40] (* _Harvey P. Dale_, Mar 05 2019 *)
%o A018921 (PARI) Vec((4-x^2-2*x^3)/((1-x)*(1-x-x^2-x^3)) + O(x^40)) \\ _Colin Barker_, Feb 13 2016
%o A018921 (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 A018921 T(4, 8, 30) \\ _Colin Barker_, Feb 14 2016
%o A018921 (Magma) Tiv:=[4,8]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..40]]; // _Bruno Berselli_, Feb 17 2016
%Y A018921 Cf. A008937.
%K A018921 nonn,easy
%O A018921 0,1
%A A018921 _R. K. Guy_
%E A018921 Comments moved to formula, and typo in data fixed by _Colin Barker_, Feb 13 2016