A010921 Shallit sequence S(3,13), a(n)=[ a(n-1)^2/a(n-2)+1 ].

3, 13, 57, 250, 1097, 4814, 21126, 92711, 406861, 1785505, 7835669, 34386747, 150905861, 662248712, 2906271193, 12754139184, 55971399613, 245629871954, 1077943993063, 4730545364606, 20759946333583, 91104796287932, 399812397069577, 1754572309731352

Offset

0,1

Comments

Matches the sequence A275634 with g.f. ( 3-2*x-2*x^2 ) / ( 1-5*x+2*x^2+3*x^3 ) for n<=9, but is then different. - R. J. Mathar, Feb 11 2016

Links

Colin Barker, Table of n, a(n) for n = 0..1000

D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

Jeffrey Shallit, Problem B-686, Fib. Quart., 29 (1991), 85.

Maple

A010921 := proc(n)
    option remember;
    if n <= 1 then
        op(n+1, [3, 13]) ;
    else
        a := procname(n-1)^2/procname(n-2) ;
        floor(1+a) ;
    end if;
end proc: # R. J. Mathar, Feb 11 2016

Mathematica

RecurrenceTable[{a[0]==3, a[1]==13, a[n]==Floor[a[n-1]^2/a[n-2]+1]}, a[n], {n, 25}] (* Harvey P. Dale, Oct 24 2011 *)

Prog

(PARI) A010921(n, a=[3, 13])={for(n=2, if(type(n)=="t_VEC", n[1], n), a=concat(a, a[n]^2\a[n-1]+1)); if(type(n)=="t_VEC", a, a[n+1])} \\ Use A010921([n]) to get the vector [a(0), ..., a(n)] \\ M. F. Hasler, Feb 11 2016
(PARI) pisotS(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1));
  a
}
pisotS(50, 3, 13) \\ Colin Barker, Aug 09 2016

Crossrefs

Cf. A008776, A275634.

Sequence in context: A151221 A020515 A049086 * A275634 A163606 A115968

Adjacent sequences: A010918 A010919 A010920 * A010922 A010923 A010924

Keyword

nonn

Author

Simon Plouffe

Status

approved