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-François 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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Definition clarified based on consultance with David Boyd by Robert Israel, Feb 12 2016
STATUS
approved