A010903 Pisot sequence E(3,13): a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134

Offset

0,1

Comments

According to Boyd (Acta Arithm. 32 (1977) p 89), quoting Pisot, every E(3,.) sequence satisfies a linear recurrence of at most order 3. Here this is easily derived from the first terms of the sequence. - R. J. Mathar, May 26 2008

A010920 coincides with this sequence for at least the first 32600 terms and probably more. - R. J. Mathar, May 26 2008

For n >= 1, a(n-1) is the number of generalized compositions of n when there are i+2 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Links

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

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.

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.

Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See pp. 11-13, 24.

Index entries for linear recurrences with constant coefficients, signature (5,-3).

Formula

a(n) = 5*a(n-1) - 3*a(n-2) = 3*A116415(n) - 2*A116415(n-1). - R. J. Mathar, May 26 2008

O.g.f.: (3-2*x)/(1-5*x+3*x^2). - R. J. Mathar, May 26 2008

a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-11+3*sqrt(13)) + (5+sqrt(13))^n*(11+3*sqrt(13))))/sqrt(13). - Colin Barker, Nov 26 2016

Mathematica

LinearRecurrence[{5, -3}, {3, 13}, 24] (* Jean-François Alcover, Oct 22 2019 *)

Prog

(PARI) Vec((3-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Jul 27 2016

Crossrefs

Cf. A010920.

Sequence in context: A081952 A234645 A010920 * A095934 A151220 A151221

Adjacent sequences: A010900 A010901 A010902 * A010904 A010905 A010906

Keyword

nonn,easy

Author

Simon Plouffe

Status

approved