A010921 - OEIS

%I #25 Jan 05 2025 19:51:34

%S 3,13,57,250,1097,4814,21126,92711,406861,1785505,7835669,34386747,

%T 150905861,662248712,2906271193,12754139184,55971399613,245629871954,

%U 1077943993063,4730545364606,20759946333583,91104796287932,399812397069577,1754572309731352

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

%C 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

%H Colin Barker, <a href="/A010921/b010921.txt">Table of n, a(n) for n = 0..1000</a>

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa34/aa3444.pdf">Some integer sequences related to the Pisot sequences</a>, Acta Arithmetica, 34 (1979), 295-305.

%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

%H Jeffrey Shallit, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/29-1/elementary29-1.pdf">Problem B-686</a>, Fib. Quart., 29 (1991), 85.

%p A010921 := proc(n)

%p     option remember;

%p     if n <= 1 then

%p         op(n+1,[3,13]) ;

%p     else

%p         a := procname(n-1)^2/procname(n-2) ;

%p         floor(1+a) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Feb 11 2016

%t 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 *)

%o (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

%o (PARI) pisotS(nmax, a1, a2) = {

%o   a=vector(nmax); a[1]=a1; a[2]=a2;

%o   for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1));

%o   a

%o }

%o pisotS(50, 3, 13) \\ _Colin Barker_, Aug 09 2016

%Y Cf. A008776, A275634.

%K nonn

%O 0,1

%A _Simon Plouffe_