%I #46 Feb 27 2024 19:09:18

%S 3,13,56,241,1037,4462,19199,82609,355448,1529413,6580721,28315366,

%T 121834667,524227237,2255632184,9705479209,41760499493,179686059838,

%U 773148800711,3326685824041,14313982718072,61589856118237,265007332436969,1140267093830134

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

%C 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

%C A010920 coincides with this sequence for at least the first 32600 terms and probably more. - _R. J. Mathar_, May 26 2008

%C 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

%H Colin Barker, <a href="/A010903/b010903.txt">Table of n, a(n) for n = 0..1000</a>

%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/1707.07798">(an + b)-color compositions</a>, arXiv:1707.07798 [math.CO], 2017.

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa34/aa3444.pdf">Some integer sequences related to the Pisot sequences</a>, Acta Arithmetica, 34 (1979), 295-305

%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae (2023). See pp. 11-13, 24.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3).

%F a(n) = 5*a(n-1) - 3*a(n-2) = 3*A116415(n) - 2*A116415(n-1). - _R. J. Mathar_, May 26 2008

%F O.g.f.: (3-2*x)/(1-5*x+3*x^2). - _R. J. Mathar_, May 26 2008

%F 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

%t LinearRecurrence[{5, -3}, {3, 13}, 24] (* _Jean-François Alcover_, Oct 22 2019 *)

%o (PARI) Vec((3-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ _Colin Barker_, Jul 27 2016

%Y Cf. A010920.

%K nonn,easy

%O 0,1

%A _Simon Plouffe_