

A010903


Pisot sequence E(3,13): a(n) = floor(a(n1)^2/a(n2) + 1/2).


3



3, 13, 56, 241, 1037, 4462, 19199, 82609, 355448, 1529413, 6580721, 28315366, 121834667, 524227237, 2255632184, 9705479209, 41760499493, 179686059838, 773148800711, 3326685824041, 14313982718072, 61589856118237, 265007332436969, 1140267093830134
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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(n1) 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, 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), 295305
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
Index entries for linear recurrences with constant coefficients, signature (5,3).


FORMULA

a(n) = 5*a(n1)  3*a(n2) = 3*A116415(n)  2*A116415(n1).  R. J. Mathar, May 26 2008
O.g.f.: (32*x)/(15*x+3*x^2).  R. J. Mathar, May 26 2008
a(n) = (2^(1n)*((5sqrt(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] (* JeanFrançois Alcover, Oct 22 2019 *)


PROG

(PARI) Vec((32*x)/(15*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



