A018918 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(3,6).

3, 6, 11, 20, 36, 64, 113, 199, 350, 615, 1080, 1896, 3328, 5841, 10251, 17990, 31571, 55404, 97228, 170624, 299425, 525455, 922110, 1618191, 2839728, 4983376, 8745216, 15346785, 26931731, 47261894, 82938843, 145547524, 255418100, 448227520, 786584465

Offset

0,1

Comments

Not to be confused with the Pisot T(3,6) sequence as defined in A008776 which is A007283. - R. J. Mathar, Feb 17 2016

Links

Colin Barker, Table of n, a(n) for n = 0..1000

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

Index entries for Pisot sequqences

Formula

Conjectures from Colin Barker, Dec 21 2012: (Start)

a(n) = 3*a(n-1)-3*a(n-2)+2*a(n-3)-a(n-4).

G.f.: -(x^3-2*x^2+3*x-3) / ((x-1)*(x^3-x^2+2*x-1)). (End)

a(n) = ceiling( a(n-1)^2/a(n-2)-1 ), by definition. - Bruno Berselli, Feb 16 2016

Mathematica

RecurrenceTable[{a[1] == 3, a[2] == 6, a[n] == Ceiling[a[n-1]^2/a[n-2] - 1]}, a, {n, 40}] (* Vincenzo Librandi, Feb 17 2016 *)

Prog

(Magma) Tiv:=[3, 6]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2)-1): n in [1..40]]; // Bruno Berselli, Feb 17 2016
(PARI) T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(3, 6, 50) \\ Colin Barker, Jul 29 2016

Crossrefs

Seems to be A010901(n) - 1 and (see conjecture) A077855(n+1).

Sequence in context: A182845 A265076 A055417 * A077855 A054887 A019302

Adjacent sequences: A018915 A018916 A018917 * A018919 A018920 A018921

Keyword

nonn

Author

R. K. Guy

Status

approved