A018922 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(8,16).

8, 16, 31, 60, 116, 224, 432, 833, 1606, 3096, 5968, 11504, 22175, 42744, 82392, 158816, 306128, 590081, 1137418, 2192444, 4226072, 8146016, 15701951, 30266484, 58340524, 112454976, 216763936, 417825921, 805385358, 1552430192, 2992405408, 5768046880

Offset

0,1

Comments

Not to be confused with the Pisot T(8,16), which is essentially A000079. - R. J. Mathar, Feb 13 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 linear recurrences with constant coefficients, signature (2,0,0,0,-1).

Index entries for Pisot sequences

Formula

a(n) = 2*a(n-1) - a(n-5).

a(n) = A107066(n+3). - Vladimir Joseph Stephan Orlovsky, Jul 08 2011

O.g.f: -(-8+x^2+2*x^3+4*x^4)/((x-1)*(x^4+x^3+x^2+x-1)) = (1/3)/(x-1)+(1/3)*(-13*x^3-20*x^2-24*x-25)/(x^4+x^3+x^2+x-1) . - R. J. Mathar, Dec 02 2007

Mathematica

Drop[CoefficientList[Series[1/(1 - 2*z + z^5), {z, 0, 100}], z], 3] (* Vladimir Joseph Stephan Orlovsky, Jul 08 2011 *)
RecurrenceTable[{a[1] == 8, a[2] == 16, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 40}] (* Bruno Berselli, Feb 17 2016 *)
LinearRecurrence[{2, 0, 0, 0, -1}, {8, 16, 31, 60, 116}, 40] (* Harvey P. Dale, Sep 21 2024 *)

Prog

(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(8, 16, 40) \\ Colin Barker, Feb 14 2016
(Magma) Tiv:=[8, 16]; [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

Crossrefs

Cf. A107066.

Sequence in context: A332918 A363280 A266086 * A290287 A303981 A285315

Adjacent sequences: A018919 A018920 A018921 * A018923 A018924 A018925

Keyword

nonn,easy

Author

R. K. Guy

Status

approved