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

0,2

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.

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

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)];

Prog

(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
(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

Crossrefs

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

Sequence in context: A368898 A326665 A100920 * A253936 A191032 A003147

Adjacent sequences: A274943 A274944 A274945 * A274947 A274948 A274949

Keyword

nonn

Author

N. J. A. Sloane, Jul 26 2016

Status

approved