A010920 Pisot sequence T(3,13), a(n) = floor( a(n-1)^2/a(n-2) ).
3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072
Offset: 0
Links
- Vincenzo Librandi, 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
- 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.
Programs
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Magma
I:=[3,13]; [n le 2 select I[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..25]]; // Bruno Berselli, Sep 03 2013
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Mathematica
RecurrenceTable[{a[0] == 3, a[1] == 13, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 25}] (* Bruno Berselli, Sep 03 2013 *)
Formula
Empirical G.f.: (3-2*x)/(1-5*x+3*x^2). - Colin Barker, Feb 21 2012
Empirical: a(n) = 5*a(n-1)-3*a(n-2) with n>1, a(0)=3 and a(1)=13. - Vincenzo Librandi, Apr 17 2012
The empirical g.f. and recurrence above hold for n<=6000. - Bruno Berselli, Sep 03 2013
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