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A022018 Define the sequence UD(a(0),a(1)) by a(n) is the least integer such that a(n)/a(n-1) > a(n-1)/a(n-2)+1 for even n >= 2 and such that a(n)/a(n-1) > a(n-1)/a(n-2) for odd n>=2. This is UD(2,16). 12

%I

%S 2,16,129,1040,8385,67604,545057,4394520,35430801,285660700,

%T 2303138321,18569044064,149712848033,1207059275044,9731910872129,

%U 78463494859944,632611632651505,5100428912583468,41122188953879473,331547494013013232,2673100425407651457

%N Define the sequence UD(a(0),a(1)) by a(n) is the least integer such that a(n)/a(n-1) > a(n-1)/a(n-2)+1 for even n >= 2 and such that a(n)/a(n-1) > a(n-1)/a(n-2) for odd n>=2. This is UD(2,16).

%C The definition uses a recurrence of Shallit's S(a0,a1) sequences if n is even and Pisot T(a0,a1) sequences if n is odd. The UD notation reflects that we are rounding up or down depending on the position in the sequence. - _David Boyd_, Feb 12 2016

%H Colin Barker, <a href="/A022018/b022018.txt">Table of n, a(n) for n = 0..1000</a>

%H D. W. Boyd, <a href="https://www.researchgate.net/publication/258834801">Linear recurrence relations for some generalized Pisot sequences</a>, Adv. Numb. Theory, Oxford Univ. Press (1991) 333-340

%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133">Linear recurrence relations for some generalized Pisot sequences</a>, (1996)

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,1,-4).

%F Empirical g.f: (2-x^2)/(1-8*x-x^2+4*x^3), holds at least up to n<=50000. - _Robert Israel_, Feb 10 2016

%F The empirical g.f. found by _Robert Israel_ has been proved. One needs only the definition and the first 6 terms of the sequence. The denominator of the g.f. is the reciprocal of a Pisot polynomial with 2nd largest root real and negative. - _David Boyd_, Mar 06 2016

%F a(n) = 8*a(n-1)+a(n-2)-4*a(n-3) for n>2. - _Colin Barker_, Aug 09 2016

%p UD := proc(a0,a1,n)

%p option remember;

%p if n = 0 then

%p a0 ;

%p elif n = 1 then

%p a1;

%p elif type(n,'even') then

%p floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)+1) ;

%p else

%p floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)) ;

%p end if;

%p end proc:

%p A022018 := proc(n)

%p UD(2,16,n) ;

%p end proc: # _R. J. Mathar_, Feb 12 2016

%t LinearRecurrence[{8, 1, -4}, {2, 16, 129}, 30] (* _Jean-Fran├žois Alcover_, Dec 12 2016 *)

%o (PARI) a=[2,16,129]; c=Colrev([8,1,-4]); for(n=2,20,a=concat(a,a[-3..-1]*c));a \\ Reproduces the data. - _M. F. Hasler_, Feb 10 2016

%o (MAGMA) Iv:=[2,16]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2))+(1-(-1)^n)/2: n in [1..20]]; // _Bruno Berselli_, Feb 11 2016

%K nonn

%O 0,1

%A _R. K. Guy_

%E Definition clarified based on consultance with _David Boyd_ by _Robert Israel_, Feb 12 2016

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)