A274946 Boyd's Pisot-like sequence F(0,5,11).

0, 5, 11, 19, 30, 46, 70, 106, 160, 241, 363, 547, 824, 1241, 1869, 2815, 4240, 6386, 9618, 14486, 21818, 32861, 49493, 74543, 112272, 169097, 254683, 383587, 577734, 870146, 1310558, 1973878, 2972928, 4477633, 6743923, 10157263, 15298216, 23041189, 34703157, 52267663, 78722192

Offset: 0

Author

N. J. A. Sloane, Jul 26 2016

Keywords

• nonn

Comments

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).

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. See page 296.

Crossrefs

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

Programs

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

Maple

f:=proc(n) option remember; global f0,f1,f2;
if n = 0 then f0
elif n=1 then f1
elif n=2 then f2
elif f(n-2)=0 then -f(n-4)
else floor(1/2 + (f(n-1)/f(n-2))*(f(n-1)+f(n-3))-f(n-2)); fi;
end;
f0:=0; f1:=5; f2:=11; [seq(f(n),n=0..40)];

PARI

boyd(nmax, f1, f2, f3) = {
  f=vector(nmax); f[1]=f1; f[2]=f2; f[3]=f3;
  for(n=4, nmax, f[n] = floor(1/2 + (f[n-1]/f[n-2])*(f[n-1]+f[n-3])-f[n-2]));
  f
}
boyd(50, 0, 5, 11) \\ Colin Barker, Jul 26 2016

Formula

Conjectures from Colin Barker, Jul 26 2016: (Start)

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

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

(End)

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