A010925 -id:A010925 - cp's OEIS frontend

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

N. J. A. Sloane, Jul 26 2016

nonn

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).

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.

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

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

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)];

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

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

A275904 Order of homogeneous linear recurrence satisfied by the Pisot sequence T(n, n^2-n+1).

Original entry on oeis.org

1, 2, 6, 36, 2048

Offset: 1

N. J. A. Sloane, Aug 11 2016

Degree of denominator of minimal g.f. for T(n, n^2-n+1).

Conjecture: a(6) = 6852224. The conjectured generating function for T(6,31) is A(x)/(1+x - x*A(x)) where A(x) = 6 + x - x^2 - x^4 - x^22 - x^1130 - x^6852224 (and as usual there is a common factor of (1+x) in numerator and denominator). - David Boyd, Aug 12 2016.

			T(1,1) is the all-ones sequence, with g.f. 1/(1-x).
T(2,3) is 2,3,4,5,6,... with g.f. (2-x)/(1-2*x+x^2).
T(3,7) is A020746, with a linear recurrence of order 6.
T(4,13) is A010919, with a linear recurrence of order 36.
T(5,21) is A010925, with a linear recurrence of order 2048.

David Boyd, Email communication to N. J. A. Sloane, Aug 06 2016

D. W. Boyd, Pisot sequences which satisfy no linear recurrences, Acta Arith. 32 (1) (1977) 89-98
D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
D. W. Boyd, On linear recurrence relations satisfied by Pisot sequences, Acta Arithm. 47 (1) (1986) 13
D. W. Boyd, Pisot sequences which satisfy no linear recurrences. II, Acta Arithm. 48 (1987) 191
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, Oxford Univ. Press, New York, 1993; with updates from 1996 and 1999.

Cf. A008776, A020746, A010919, A010925.

cp's OEIS Frontend

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

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