%I #49 Jul 13 2023 09:29:46

%S 3,9,26,75,216,622,1791,5157,14849,42756,123111,354484,1020696,

%T 2938977,8462447,24366645,70160958,202020427,581694636,1674922950,

%U 4822748423,13886550633,39984728949,115131438424,331507764639,954538564968

%N 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(3,9).

%C Let M denotes the 4 X 4 matrix = row by row (1,1,1,1)(1,1,1,0)(1,1,0,0)(1,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n))=M^n*A where A is the vector (1,1,1,1) then a(n)=y(n+1). - _Benoit Cloitre_, Apr 02 2002

%C Not to be confused with the Pisot T(3,9) sequence, which is A000244. - _R. J. Mathar_, Feb 13 2016

%H Vincenzo Librandi, <a href="/A018919/b018919.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-1).

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

%F For n>1, a(n) = ceiling(a(n-1)^2/a(n-2)) - 1.

%F For n>2, a(n) = 3*a(n-1) - a(n-3).

%F G.f.: -(x^2-3) / (x^3-3*x+1). - _Colin Barker_, Dec 13 2012

%t CoefficientList[Series[- (x^2 - 3)/(x^3 - 3 x + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 16 2013 *)

%t RecurrenceTable[{a[1] == 3, a[2] == 9, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 30}] (* _Bruno Berselli_, Feb 17 2016 *)

%t LinearRecurrence[{3,0,-1},{3,9,26},30] (* _Harvey P. Dale_, Feb 06 2019 *)

%o (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 T(3, 9, 30) \\ _Colin Barker_, Feb 14 2016

%o (Magma) Tiv:=[3,9]; [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 Cf. A076264.

%K nonn,easy

%O 0,1

%A _R. K. Guy_