%I #25 Sep 08 2022 08:46:17

%S 0,5,11,19,30,46,70,106,160,241,363,547,824,1241,1869,2815,4240,6386,

%T 9618,14486,21818,32861,49493,74543,112272,169097,254683,383587,

%U 577734,870146,1310558,1973878,2972928,4477633,6743923,10157263,15298216,23041189,34703157,52267663,78722192

%N Boyd's Pisot-like sequence F(0,5,11).

%C F(f0,f1,f2) is the sequence a(n) defined by a(0)=f0, a(1)=f1, a(2)=f2, and for n >= 3, a(n) = floor(1/2 + (a(n-1)/a(n-2))*(a(n-1)+a(n-3))-a(n-2)) unless a(n-2)=0 in which case a(n) = - a(n-4).

%H Colin Barker, <a href="/A274946/b274946.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. See page 296.

%F Conjectures from _Colin Barker_, Jul 26 2016: (Start)

%F a(n) = a(n-1)+a(n-3)+a(n-5)-a(n-6) for n>5.

%F G.f.: x*(5+6*x+8*x^2+6*x^3+5*x^4) / (1-x-x^3-x^5+x^6).

%F (End)

%F Note the warning in A010925 from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004: [A010925] and other examples show that it is essential to reject conjectured generating functions for Pisot sequences until a proof or reference is provided. - _N. J. A. Sloane_, Jul 26 2016

%p f:=proc(n) option remember; global f0,f1,f2;

%p if n = 0 then f0

%p elif n=1 then f1

%p elif n=2 then f2

%p elif f(n-2)=0 then -f(n-4)

%p else floor(1/2 + (f(n-1)/f(n-2))*(f(n-1)+f(n-3))-f(n-2)); fi;

%p end;

%p f0:=0; f1:=5; f2:=11; [seq(f(n),n=0..40)];

%o (Magma) f:=[0,5,11]; [n le 3 select f[n] else Floor(1/2+(Self(n-1)/Self(n-2))*(Self(n-1)+Self(n-3))-Self(n-2)): n in [1..50]]; // _Bruno Berselli_, Jul 26 2016

%o (PARI) boyd(nmax, f1, f2, f3) = {

%o f=vector(nmax); f[1]=f1; f[2]=f2; f[3]=f3;

%o for(n=4, nmax, f[n] = floor(1/2 + (f[n-1]/f[n-2])*(f[n-1]+f[n-3])-f[n-2]));

%o f

%o }

%o boyd(50, 0, 5, 11) \\ _Colin Barker_, Jul 26 2016

%Y Cf. A008776 (definition of the usual Pisot sequences), A010925.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jul 26 2016