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A015523 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=0, a(1)=1.

%I A015523 #55 Aug 01 2022 08:04:56
%S A015523 0,1,3,14,57,241,1008,4229,17727,74326,311613,1306469,5477472,
%T A015523 22964761,96281643,403668734,1692414417,7095586921,29748832848,
%U A015523 124724433149,522917463687,2192374556806,9191710988853,38537005750589
%N A015523 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=0, a(1)=1.
%C A015523 From _Johannes W. Meijer_, Aug 01 2010: (Start)
%C A015523 a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 and 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
%C A015523 For n >= 1, the sequence above corresponds to 24 red king vectors, i.e., A[5] vectors, with decimal values 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the side squares to A152187 and for the central square to A179606.
%C A015523 This sequence belongs to a family of sequences with g.f. 1/(1-3*x-k*x^2). Red king sequences that are members of this family are A007482 (k=2), A015521 (k=4), A015523 (k=5; this sequence), A083858 (k=6), A015524 (k=7) and A015525 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A049072 (k=-4), A057083 (k=-3), A000225 (k=-2), A001906 (k=-1), A000244 (k=0), A006190 (k=1), A030195 (k=3), A099012 (k=9), A015528 (k=10) and A015529 (k=11).
%C A015523 Inverse binomial transform of A052918 (with extra leading 0).
%C A015523 (End)
%C A015523 First differences in A197189. - _Bruno Berselli_, Oct 11 2011
%C A015523 Pisano period lengths:  1, 3, 4, 6, 4, 12, 3, 12, 12, 12, 120, 12, 12, 3, 4, 24, 288, 12, 72, 12, ... - _R. J. Mathar_, Aug 10 2012
%C A015523 This is the Lucas U(P=3, Q=-5) sequence, and hence for n >= 0, a(n+2)/a(n+1) equals the continued fraction 3 + 5/(3 + 5/(3 + 5/(3 + ... + 5/3))) with n 5's. - _Greg Dresden_, Oct 06 2019
%H A015523 Vincenzo Librandi, <a href="/A015523/b015523.txt">Table of n, a(n) for n = 0..1000</a>
%H A015523 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,5).
%F A015523 a(n) = 3*a(n-1) + 5*a(n-2).
%F A015523 From _Paul Barry_, Jul 20 2004: (Start)
%F A015523 a(n) = ((3/2 + sqrt(29)/2)^n - (3/2 - sqrt(29)/2)^n)/sqrt(29).
%F A015523 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1,k)*5^k*3^(n-2*k-1). (End)
%F A015523 G.f.: x/(1 - 3*x - 5*x^2). - _R. J. Mathar_, Nov 16 2007
%F A015523 From _Johannes W. Meijer_, Aug 01 2010: (Start)
%F A015523 Limit_{k->oo} a(n+k)/a(k) = (A072263(n) + a(n)*sqrt(29))/2.
%F A015523 Limit_{n->oo} A072263(n)/a(n) = sqrt(29). (End)
%F A015523 G.f.: G(0)*x/(2-3*x), where G(k) = 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 17 2013
%F A015523 E.g.f.: 2*exp(3*x/2)*sinh(sqrt(29)*x/2)/sqrt(29). - _Stefano Spezia_, Oct 06 2019
%t A015523 Join[{a = 0, b = 1}, Table[c = 3 * b + 5 * a; a = b; b = c, {n, 100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)
%t A015523 a[0] := 0; a[1] := 1; a[n_] := a[n] = 3a[n - 1] + 5a[n - 2]; Table[a[n], {n, 0, 49}] (* _Alonso del Arte_, Jan 16 2011 *)
%o A015523 (Sage) [lucas_number1(n,3,-5) for n in range(0, 24)] # _Zerinvary Lajos_, Apr 22 2009
%o A015523 (Magma) [ n eq 1 select 0 else n eq 2 select 1 else 3*Self(n-1)+5*Self(n-2): n in [1..30] ]; // _Vincenzo Librandi_, Aug 23 2011
%o A015523 (PARI) x='x+O('x^30); concat([0], Vec(x/(1-3*x-5*x^2))) \\ _G. C. Greubel_, Jan 01 2018
%Y A015523 Cf. A072263, A072264, A152187, A179606, A197189.
%K A015523 nonn,easy
%O A015523 0,3
%A A015523 _Olivier Gérard_