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

1, 6, 35, 205, 1200, 7025, 41125, 240750, 1409375, 8250625, 48300000, 282753125, 1655265625, 9690093750, 56726796875, 332084453125, 1944056250000, 11380703515625, 66623798828125, 390022511718750, 2283231552734375

Offset

0,2

Comments

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.

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.

Inverse binomial transform of A004187 (without the leading 0).

Equals the INVERT transform of A086347 and the INVERTi transform of A180167. - Gary W. Adamson, Aug 14 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 7.

Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Index entries for linear recurrences with constant coefficients, signature (5,5).

Formula

G.f.: (1+x)/(1 - 5*x - 5*x^2).

a(n) = 5*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 6.

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.

Lim_{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).

Lim_{k->oo} a(2*n+k)/a(k) = 2*A000351(n)/(A056854(n) - 3*A004187(n)*sqrt(5)) for n >= 1.

Lim_{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.

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

Maple

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);

Mathematica

LinearRecurrence[{5, 5}, {1, 6}, 30] (* Vincenzo Librandi, Nov 15 2011 *)

Prog

(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
(PARI) my(x='x+O('x^30)); Vec((1+x)/(1-5*x-5*x^2)) \\ G. C. Greubel, Apr 07 2019
(Sage) ((1+x)/(1-5*x-5*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 07 2019

Crossrefs

Cf. A057088, A180032.

Cf. A086347, A180167. - Gary W. Adamson, Aug 14 2010

Sequence in context: A242629 A001109 A352972 * A354134 A260770 A262717

Adjacent sequences: A180030 A180031 A180032 * A180034 A180035 A180036

Keyword

easy,nonn

Author

Johannes W. Meijer, Aug 09 2010

Status

approved