A202395 -id:A202395 - cp's OEIS frontend

A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2x)/(1 - 5x - 3*x^2).

1, 3, 18, 99, 549, 3042, 16857, 93411, 517626, 2868363, 15894693, 88078554, 488076849, 2704619907, 14987330082, 83050510131, 460214540901, 2550224234898, 14131764797193, 78309496690659, 433942777844874

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) 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 A180028.

The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values varying between 7 and 448. The corner and side squares lead for these vectors to A180035.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (5, 3).

I:=[1,3]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0,0,0,0,0,0,1,1,1]: 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);

LinearRecurrence[{5,3},{1,3},201] (* Vincenzo Librandi, Nov 15 2011 *)

G.f.: (1-2*x)/(1 - 5*x - 3*x^2).
a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 3.
a(n) = ((1+16*A)*A^(-n-1) + (1+16*B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.
a(n) = Sum_{k=0..n} A202395(n,k)*2^k. - Philippe Deléham, Dec 21 2011

Second formula corrected by Vincenzo Librandi, Nov 15 2011

A202396 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 13, 27, 19, 5, 34, 86, 86, 42, 8, 89, 265, 338, 234, 85, 13, 233, 798, 1227, 1084, 567, 166, 21, 610, 2362, 4230, 4510, 3038, 1286, 314, 34, 1597, 6898, 14058, 17474, 14284, 7814, 2774, 582, 55

Offset: 0

Philippe Deléham, Dec 18 2011

T(n,n) = Fibonacci(n+2) = A000045(n+2).

			Triangle begins :
1
2, 2
5, 8, 3
13, 27, 19, 5
34, 86, 86, 42, 8
89, 265, 338, 234, 85, 13

Cf. A000045, A001519, A122367, A000302 (row sums), A202395

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if n

G.f.: (1+(y-1)*x)/(1-(3+y)*x+(1-y^2)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A122367(n), A000302(n), A180035(n) for x = -1, 0, 1, 2 respectively.

Sum_{k, 0<=k<=n} T(n,k)*3^k = 2^n * A055099(n). - Philippe Deléham, Feb 05 2012

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cp's OEIS Frontend

A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2x)/(1 - 5x - 3*x^2).

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A202396 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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cp's OEIS Frontend

A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 5*x - 3*x^2).

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Magma

Maple

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Extensions

A202396 Triangle T(n,k), read by rows, given by (2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A180036 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2x)/(1 - 5x - 3*x^2).