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A123011 a(n) = 2*a(n-1) + 5*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

%I A123011 #19 Sep 15 2024 13:43:03
%S A123011 1,5,15,55,185,645,2215,7655,26385,91045,314015,1083255,3736585,
%T A123011 12889445,44461815,153370855,529050785,1824955845,6295165615,
%U A123011 21715110455,74906048985,258387650245,891305545415,3074549342055
%N A123011 a(n) = 2*a(n-1) + 5*a(n-2) for n > 1; a(0) = 1, a(1) = 5.
%H A123011 G. C. Greubel, <a href="/A123011/b123011.txt">Table of n, a(n) for n = 0..1000</a>
%H A123011 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,5).
%F A123011 a(n) = ((3+2*sqrt(6))*(1+sqrt(6))^n + (3-2*sqrt(6))*(1-sqrt(6))^n)/6. - _Klaus Brockhaus_, Aug 15 2009
%F A123011 From _Klaus Brockhaus_, Aug 15 2009: (Start)
%F A123011 G.f.: (1+3*x)/(1-2*x-5*x^2).
%F A123011 Binomial transform of A164532.
%F A123011 Inverse binomial transform of A164549. (End)
%F A123011 a(n) = (sqrt(5)*i)^(n-1)*(sqrt(5)*i*ChebyshevU(n, -i/sqrt(5)) + 3*ChebyshevU(n-1, -i/sqrt(5))) for n > 0 with a(0) = 1. - _G. C. Greubel_, Jul 13 2021
%t A123011 LinearRecurrence[{2,5}, {1,5}, 31] (* _G. C. Greubel_, Jul 13 2021 *)
%o A123011 (Magma) [ n le 2 select 4*n-3 else 2*Self(n-1)+5*Self(n-2): n in [1..24] ]; // _Klaus Brockhaus_, Aug 15 2009
%o A123011 (Sage) [1]+[(sqrt(5)*i)^(n-1)*(sqrt(5)*i*chebyshev_U(n, -i/sqrt(5)) + 3*chebyshev_U(n-1, -i/sqrt(5))) for n in (1..30)] # _G. C. Greubel_, Jul 13 2021
%Y A123011 Cf. A002532, A164532, A164549.
%K A123011 nonn
%O A123011 0,2
%A A123011 _Roger L. Bagula_, Sep 23 2006
%E A123011 Edited by _N. J. A. Sloane_, Aug 27 2009, using simpler definition suggested by _Klaus Brockhaus_, Aug 15 2009