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A021006 Pisot sequence P(4,11), a(0)=4, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1). Evidently satisfies a(n) = 2*a(n-1)+2*a(n-2).

%I A021006 #50 Jan 04 2025 10:08:54
%S A021006 4,11,30,82,224,612,1672,4568,12480,34096,93152,254496,695296,1899584,
%T A021006 5189760,14178688,38736896,105831168,289136128,789934592,2158141440,
%U A021006 5896152064,16108587008,44009478144,120236130304,328491216896,897454694400,2451891822592,6698693033984,18301169713152,49999725494272,136601790414848
%N A021006 Pisot sequence P(4,11), a(0)=4, a(1)=11, a(n+1) is the nearest integer to a(n)^2/a(n-1). Evidently satisfies a(n) = 2*a(n-1)+2*a(n-2).
%C A021006 Pisano period lengths: 1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1,144, 9,180, 24,,.. - _R. J. Mathar_, Aug 10 2012
%C A021006 Inverse binomial transform of A001353 without its first two terms, and downshift. - _Richard R. Forberg_, Aug 24 2013
%H A021006 Vincenzo Librandi, <a href="/A021006/b021006.txt">Table of n, a(n) for n = 0..200</a>
%H A021006 Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A021006 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A021006 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,2).
%H A021006 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%F A021006 G.f.: (4+3*x)/(1-2*x-2*x^2). [_Philippe Deléham_, Nov 19 2008]
%F A021006 a(n) = (sqrt(3)/24)*((1+sqrt(3))^(n+4) - (1-sqrt(3))^(n+4)). - _Taras Goy_, Jan 04 2025
%t A021006 LinearRecurrence[{2,2},{4,11},30] (* _Harvey P. Dale_, Oct 25 2011 *)
%o A021006 (Magma) I:=[4,11]; [n le 2 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 26 2011
%K A021006 nonn,easy
%O A021006 0,1
%A A021006 _R. K. Guy_