A020745 Pisot sequence T(3,5).

3, 5, 8, 12, 18, 27, 40, 59, 87, 128, 188, 276, 405, 594, 871, 1277, 1872, 2744, 4022, 5895, 8640, 12663, 18559, 27200, 39864, 58424, 85625, 125490, 183915, 269541, 395032, 578948, 848490, 1243523, 1822472, 2670963, 3914487, 5736960, 8407924, 12322412, 18059373

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 877

Crossrefs

See A008776 for definitions of Pisot sequences.

Programs

Magma

Iv:=[3,5]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..50]]; // Bruno Berselli, Feb 04 2016

Mathematica

RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 50}] (* Bruno Berselli, Feb 04 2016 *)

Formula

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).

Empirical g.f.: (3-x+x^2-2*x^3)/(1-x)/(1-x-x^3). [Colin Barker, Feb 19 2012]