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).

2, 16, 129, 1040, 8385, 67604, 545057, 4394520, 35430801, 285660700, 2303138321, 18569044064, 149712848033, 1207059275044, 9731910872129, 78463494859944, 632611632651505, 5100428912583468, 41122188953879473, 331547494013013232, 2673100425407651457

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Adv. Numb. Theory, Oxford Univ. Press (1991) 333-340
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, (1996)
• Index entries for linear recurrences with constant coefficients, signature (8,1,-4).
• Index entries for Pisot sequences

Programs

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

Maple

UD := proc(a0,a1,n)
    option remember;
    if n = 0 then
        a0 ;
    elif n = 1 then
        a1;
    elif type(n,'even') then
        floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)+1) ;
    else
        floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)) ;
    end if;
end proc:
A022018 := proc(n)
    UD(2,16,n) ;
end proc: # R. J. Mathar, Feb 12 2016

Mathematica

LinearRecurrence[{8, 1, -4}, {2, 16, 129}, 30] (* Jean-François Alcover, Dec 12 2016 *)

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

Formula

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

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

a(n) = 8*a(n-1)+a(n-2)-4*a(n-3) for n>2. - Colin Barker, Aug 09 2016

Extensions

Definition clarified based on consultance with David Boyd by Robert Israel, Feb 12 2016