cp's OEIS Frontend

A180033 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 5*x - 5*x^2).

%I A180033 #37 Mar 17 2025 22:13:42
%S A180033 1,6,35,205,1200,7025,41125,240750,1409375,8250625,48300000,282753125,
%T A180033 1655265625,9690093750,56726796875,332084453125,1944056250000,
%U A180033 11380703515625,66623798828125,390022511718750,2283231552734375
%N A180033 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 5*x - 5*x^2).
%C A180033 The a(n) represent the number of n-move routes of a fairy chess piece starting in the corner and side squares (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032.
%C A180033 The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values between 47 and 488. The central squares lead for these vectors to A057088.
%C A180033 Inverse binomial transform of A004187 (without the leading 0).
%C A180033 Equals the INVERT transform of A086347 and the INVERTi transform of A180167. - _Gary W. Adamson_, Aug 14 2010
%H A180033 Vincenzo Librandi, <a href="/A180033/b180033.txt">Table of n, a(n) for n = 0..200</a>
%H A180033 Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, <a href="https://arxiv.org/abs/2401.13524">Combinatorics on words and generating Dirichlet series of automatic sequences</a>, arXiv:2401.13524 [math.CO], 2025. See p. 14.
%H A180033 D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.
%H A180033 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A180033 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,5).
%F A180033 G.f.: (1+x)/(1 - 5*x - 5*x^2).
%F A180033 a(n) = 5*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 6.
%F A180033 a(n) = ((7+5*A)*A^(-n-1) + (7+5*B)*B^(-n-1))/45 with A = (-5+3*sqrt(5))/10 and B = (-5-3*sqrt(5))/10.
%F A180033 Limit_{k->oo} a(n+k)/a(k) = 2*5^(n/2)/(L(2*n) - F(2*n)*sqrt(5)) with L(n) = A000032(n) and F(n) = A000045(n).
%F A180033 Limit_{k->oo} a(2*n+k)/a(k) = 2*A000351(n)/(A056854(n) - 3*A004187(n)*sqrt(5)) for n >= 1.
%F A180033 Limit_{k->oo} a(2*n-1+k)/a(k) = 2*A000351(n)/(3*A049685(n-1)*sqrt(5) - 5*A033890(n-1)) for n >= 1.
%F A180033 a(n) = A057088(n+1)/5. a(2*n) = 5^n*F(4*(n+1))/3, a(2*n+1) = 5^n*L(2*(2*n+3))/3. - _Ehren Metcalfe_, Apr 04 2019
%F A180033 E.g.f.: exp(5*x/2)*(15*cosh(3*sqrt(5)*x/2) + 7*sqrt(5)*sinh(3*sqrt(5)*x/2))/15. - _Stefano Spezia_, Mar 17 2025
%p A180033 with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [0,0,0,1,0,1,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);
%t A180033 LinearRecurrence[{5,5},{1,6}, 30] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180033 (Magma) I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%o A180033 (PARI) my(x='x+O('x^30)); Vec((1+x)/(1-5*x-5*x^2)) \\ _G. C. Greubel_, Apr 07 2019
%o A180033 (Sage) ((1+x)/(1-5*x-5*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 07 2019
%Y A180033 Cf. A057088, A180032.
%Y A180033 Cf. A086347, A180167. - _Gary W. Adamson_, Aug 14 2010
%K A180033 easy,nonn
%O A180033 0,2
%A A180033 _Johannes W. Meijer_, Aug 09 2010