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A048625 Pisot sequence P(4,6).

%I A048625 #38 Oct 25 2023 08:26:28
%S A048625 4,6,9,13,19,28,41,60,88,129,189,277,406,595,872,1278,1873,2745,4023,
%T A048625 5896,8641,12664,18560,27201,39865,58425,85626,125491,183916,269542,
%U A048625 395033,578949,848491,1243524,1822473,2670964,3914488,5736961,8407925,12322413,18059374
%N A048625 Pisot sequence P(4,6).
%C A048625 Conjecture: satisfies a linear recurrence having signature (1, 0, 1). - _Harvey P. Dale_, Jun 05 2021
%H A048625 Colin Barker, <a href="/A048625/b048625.txt">Table of n, a(n) for n = 0..1000</a>
%H A048625 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%F A048625 a(n) = a(n-1) + a(n-3) (Checked up to n = 48000).
%F A048625 G.f.: (conjecture) (( Q(0)-1)/2 -(x+x^2+x^3+2*x^4+3*x^5))/x^6, where Q(k) = 1 + x^3 + (2*k+3)*x - x*(2*k+1 + x^2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 05 2013
%p A048625 P := proc(a0,a1,n)
%p A048625     option remember;
%p A048625     if n = 0 then
%p A048625         a0 ;
%p A048625     elif n = 1 then
%p A048625         a1;
%p A048625     else
%p A048625         ceil( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)-1/2) ;
%p A048625     end if;
%p A048625 end proc:
%p A048625 A048625 := proc(n)
%p A048625     P(4,6,n) ;
%p A048625 end proc: # _R. J. Mathar_, Feb 12 2016
%t A048625 P[a0_, a1_, n_] := P[a0, a1, n] = Switch[n, 0, a0, 1, a1, _, Ceiling[P[a0, a1, n-1]^2/P[a0, a1, n-2] - 1/2]];
%t A048625 a[n_] := P[4, 6, n];
%t A048625 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Oct 25 2023, after _R. J. Mathar_ *)
%o A048625 (PARI) pisotP(nmax, a1, a2) = {
%o A048625   a=vector(nmax); a[1]=a1; a[2]=a2;
%o A048625   for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
%o A048625   a
%o A048625 }
%o A048625 pisotP(50, 4, 6) \\ _Colin Barker_, Aug 08 2016
%Y A048625 Subsequence of A000930. See A008776 for definitions of Pisot sequences.
%K A048625 nonn
%O A048625 0,1
%A A048625 _David W. Wilson_