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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
2, 16, 129, 1040, 8385, 67604, 545057, 4394520, 35430801, 285660700, 2303138321, 18569044064, 149712848033, 1207059275044, 9731910872129, 78463494859944, 632611632651505, 5100428912583468, 41122188953879473, 331547494013013232, 2673100425407651457 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

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

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-Fran├žois Alcover, Dec 12 2016 *)

PROG

(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

(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

CROSSREFS

Sequence in context: A012463 A037517 A037720 * A067684 A074623 A323946

Adjacent sequences:  A022015 A022016 A022017 * A022019 A022020 A022021

KEYWORD

nonn

AUTHOR

R. K. Guy

EXTENSIONS

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

STATUS

approved

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Last modified July 17 14:39 EDT 2019. Contains 325106 sequences. (Running on oeis4.)