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%I #25 Sep 08 2022 08:45:46
%S 2,11,58,304,1592,8336,43648,228544,1196672,6265856,32808448,
%T 171787264,899489792,4709789696,24660779008,129125515264,676109975552,
%U 3540157792256,18536506851328,97058409938944,508204432228352
%N a(n) = ((2+sqrt(5))*(3+sqrt(5))^n + (2-sqrt(5))*(3-sqrt(5))^n)/2.
%C Binomial transform of A001077 without initial 1. Third binomial transform of A162963. Inverse binomial transform of A162772.
%H Vincenzo Librandi, <a href="/A162771/b162771.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-4).
%F a(n) = 6*a(n-1) - 4*a(n-2) for n > 1; a(0) = 2, a(1) = 11. [corrected by _Harvey P. Dale_, Aug 15 2013]
%F G.f.: (2-x)/(1-6*x+4*x^2).
%F a(n) = 2^(n-1) * A002878(n+1). - _Diego Rattaggi_, Jun 16 2020
%F a(n) = Sum_{k>=1} binomial(k+n-1,n) * A000032(k) / 2^(k+1). - _Diego Rattaggi_, Aug 02 2020
%t LinearRecurrence[{6,-4},{2,11},30] (* _Harvey P. Dale_, Aug 15 2013 *)
%t CoefficientList[Series[(2 - x) / (1 - 6 x + 4 x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 16 2013 *)
%o (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((2+r)*(3+r)^n+(2-r)*(3-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Jul 19 2009
%Y Cf. A001077, A002878, A162963, A162772.
%K nonn
%O 0,1
%A Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009
%E Edited and extended beyond a(5) by _Klaus Brockhaus_, Jul 19 2009
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