|
%I #34 Sep 08 2022 08:44:40
%S 0,1,3,19,87,451,2223,11179,55767,279091,1394943,6975739,34876647,
%T 174387331,871928463,4359658699,21798260727,108991369171,544956714783,
%U 2724783836059,13623918656007,68119594328611,340597969545903
%N a(n) = 3*a(n-1) + 10*a(n-2).
%C The ratio a(n+1)/a(n) converges to 5 as n approaches infinity. - _Felix P. Muga II_, Mar 10 2014
%H Vincenzo Librandi, <a href="/A015528/b015528.txt">Table of n, a(n) for n = 0..1000</a>
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 10).
%F a(n) = 3*a(n-1) + 10*a(n-2).
%F a(n) = (5^n - (-2)^n)/7. Binomial transform is A015540. - _Paul Barry_, Feb 07 2004
%F G.f.: x/(1-x*(10*x+3)). - _Harvey P. Dale_, Jan 27 2012
%t Join[{a=0,b=1},Table[c=3*b+10*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)
%t LinearRecurrence[{3,10},{0,1},30] (* or *) CoefficientList[Series[x/(1-x (10x+3)),{x,0,29}],x] (* _Harvey P. Dale_, Jan 27 2012 *)
%o (Sage) [lucas_number1(n,3,-10) for n in range(0, 23)]# _Zerinvary Lajos_, Apr 22 2009
%o (Magma) [5^n/7-(-2)^n/7: n in [0..30]]; // _Vincenzo Librandi_, Aug 23 2011
%o (PARI) for(n=0,30, print1((5^n - (-2)^n)/7, ", ")) \\ _G. C. Greubel_, Jan 01 2018
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
|
