%I #24 Jul 21 2023 04:21:41
%S 0,1,5,35,225,1475,9625,62875,410625,2681875,17515625,114396875,
%T 747140625,4879671875,31869765625,208145546875,1359425390625,
%U 8878582421875,57987166015625,378721654296875,2473479931640625,16154616201171875,105507880322265625
%N a(n) = 5*a(n-1) + 10*a(n-2), with a(1)=0 and a(2)=1.
%H Nathaniel Johnston, <a href="/A180250/b180250.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,10).
%F a(n) = ((5+sqrt(65))^(n-1) - (5-sqrt(65))^(n-1))/(2^(n-1)*sqrt(65)). - _Rolf Pleisch_, May 14 2011
%F G.f.: x^2/(1-5*x-10*x^2).
%F a(n) = (i*sqrt(10))^(n-1) * ChebyshevU(n-1, -i*sqrt(5/8)). - _G. C. Greubel_, Jul 21 2023
%t Join[{a=0,b=1},Table[c=5*b+10*a;a=b;b=c,{n,100}]]
%t LinearRecurrence[{5,10}, {0,1}, 30] (* _G. C. Greubel_, Jan 16 2018 *)
%o (PARI) a(n)=([0,1;10,5]^(n-1))[1,2] \\ _Charles R Greathouse IV_, Oct 03 2016
%o (PARI) my(x='x+O('x^30)); concat([0], Vec(x^2/(1-5*x-10*x^2))) \\ _G. C. Greubel_, Jan 16 2018
%o (Magma) [n le 2 select n-1 else 5*Self(n-1) +10*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 16 2018
%o (SageMath)
%o A180250= BinaryRecurrenceSequence(5,10,0,1)
%o [A180250(n-1) for n in range(1,41)] # _G. C. Greubel_, Jul 21 2023
%Y Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.
%K nonn,easy
%O 1,3
%A _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011