cp's OEIS Frontend

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).

%I A022018 #59 Jul 13 2023 09:45:43
%S A022018 2,16,129,1040,8385,67604,545057,4394520,35430801,285660700,
%T A022018 2303138321,18569044064,149712848033,1207059275044,9731910872129,
%U A022018 78463494859944,632611632651505,5100428912583468,41122188953879473,331547494013013232,2673100425407651457
%N 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).
%C A022018 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 A022018 Colin Barker, <a href="/A022018/b022018.txt">Table of n, a(n) for n = 0..1000</a>
%H A022018 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 A022018 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 A022018 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,1,-4).
%H A022018 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%F A022018 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 A022018 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 A022018 a(n) = 8*a(n-1)+a(n-2)-4*a(n-3) for n>2. - _Colin Barker_, Aug 09 2016
%p A022018 UD := proc(a0,a1,n)
%p A022018     option remember;
%p A022018     if n = 0 then
%p A022018         a0 ;
%p A022018     elif n = 1 then
%p A022018         a1;
%p A022018     elif type(n,'even') then
%p A022018         floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)+1) ;
%p A022018     else
%p A022018         floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)) ;
%p A022018     end if;
%p A022018 end proc:
%p A022018 A022018 := proc(n)
%p A022018     UD(2,16,n) ;
%p A022018 end proc: # _R. J. Mathar_, Feb 12 2016
%t A022018 LinearRecurrence[{8, 1, -4}, {2, 16, 129}, 30] (* _Jean-François Alcover_, Dec 12 2016 *)
%o A022018 (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 A022018 (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 A022018 nonn,easy
%O A022018 0,1
%A A022018 _R. K. Guy_
%E A022018 Definition clarified based on consultance with _David Boyd_ by _Robert Israel_, Feb 12 2016