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A245561 a(n) = 5^n - ( (sqrt(5)*phi)^n + (sqrt(5)/phi)^n ) + 1, where phi = golden ratio A001622.

%I A245561 #17 Sep 08 2022 08:46:09
%S A245561 0,1,11,76,451,2501,13376,70001,361251,1846876,9381251,47437501,
%T A245561 239109376,1202500001,6037656251,30279296876,151725781251,
%U A245561 759820312501,3803412109376,19032656250001,95219707031251,476302685546876,2382252050781251,11913932617187501
%N A245561 a(n) = 5^n - ( (sqrt(5)*phi)^n + (sqrt(5)/phi)^n ) + 1, where phi = golden ratio A001622.
%D A245561 Roger L. Bagula, email message, Aug 08 2014.
%H A245561 Vincenzo Librandi, <a href="/A245561/b245561.txt">Table of n, a(n) for n = 0..200</a>
%H A245561 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (11,-40,55,-25).
%F A245561 a(n) = 5^n - A020876(n) + 1.
%F A245561 G.f.: -x*(5*x^2-1)/((1-5*x+5*x^2)*(x-1)*(5*x-1)). -  _Vincenzo Librandi_, Aug 08 2014
%p A245561 g:=n->simplify(rationalize(simplify(expand( (sqrt(5)*p)^n + (sqrt(5)*q)^n ))); # A020876
%p A245561 h:=n->5^n-g(n)+1;
%p A245561 [seq(h(n),n=0..40)];
%t A245561 CoefficientList[Series[-x (5 x^2 - 1)/((1 - 5 x + 5 x^2) (x - 1)(5 x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 08 2014 *)
%t A245561 LinearRecurrence[{11,-40,55,-25},{0,1,11,76},30] (* _Harvey P. Dale_, Nov 05 2017 *)
%o A245561 (Magma) [5^n + 1 - Floor(((5+Sqrt(5))/2)^n+((5-Sqrt(5))/2)^n): n in [0..30]]; // _Vincenzo Librandi_, Aug 08 2014
%Y A245561 Cf. A020876, A001622.
%K A245561 nonn
%O A245561 0,3
%A A245561 _N. J. A. Sloane_, Aug 08 2014