This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A152187 #28 Jan 03 2024 08:45:44
%S A152187 1,5,20,85,355,1490,6245,26185,109780,460265,1929695,8090410,33919705,
%T A152187 142211165,596232020,2499751885,10480415755,43940006690,184222098845,
%U A152187 772366329985,3238209484180,13576460102465,56920427728295
%N A152187 a(n) = 3*a(n-1) + 5*a(n-2), with a(0)=1, a(1)=5.
%C A152187 Unsigned version of A152185.
%C A152187 From _Johannes W. Meijer_, Aug 01 2010: (Start)
%C A152187 The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 and 8) 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 A152187 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 corner squares to A015523 and for the central square to A179606.
%C A152187 This sequence belongs to a family of sequences with g.f. (1+2*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k=2), A108981 (k=4), A152187 (k=5; this sequence), A154964 (k=6), A179602 (k=7) and A179598 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A036563 (k=-2), A054486 (k=-1), A084244 (k=0), A108300 (k=1) and A000351 (k=10).
%C A152187 Inverse binomial transform of A015449 (without the first leading 1).
%C A152187 (End)
%H A152187 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 5).
%F A152187 G.f.: (1+2*x)/(1 - 3*x - 5*x^2).
%F A152187 Lim_{k->infinity} a(n+k)/a(k) = (A072263(n) + A015523(n)*sqrt(29))/2. - _Johannes W. Meijer_, Aug 01 2010
%F A152187 G.f.: G(0)*(1+2*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
%t A152187 LinearRecurrence[{3,5},{1,5},40] (* _Harvey P. Dale_, May 03 2013 *)
%Y A152187 Cf. A015523, A072263, A072264, A179606, A197189.
%K A152187 nonn,easy
%O A152187 0,2
%A A152187 _Philippe Deléham_, Nov 28 2008