A041006 -id:A041006 - cp's OEIS frontend

A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 10x^2 - 4*x^3).

1, 3, 16, 66, 304, 1332, 5968, 26472, 117952, 524496, 2334400, 10385568, 46213120, 205619520, 914912512, 4070872704, 18113348608, 80595074304, 358607125504, 1595618388480, 7099688329216, 31589989045248, 140559334936576

Offset: 0

Johannes W. Meijer, Jul 28 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 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.

The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the side squares to A123347 and for the central square to A179601.

Index entries for linear recurrences with constant coefficients, signature (2,10,4).

with(LinearAlgebra): nmax:=24; m:=1; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,1,1,0,0,0,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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);

Vec((1+x)/(1 - 2*x - 10*x^2 - 4*x^3) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

G.f.: (1+x)/(1 - 2*x - 10*x^2 - 4*x^3).
a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=16.
a(n) = (4*(-1/2)^(-n) + (1+sqrt(6))*A^(-n-1) + (1-sqrt(6))*B^(-n-1))/20 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1))) for n => 1.

A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4x)/(1 - 2x - 10x^2 - 4x^3).

Original entry on oeis.org

1, 6, 22, 108, 460, 2088, 9208, 41136, 182704, 813600, 3618784, 16104384, 71651008, 318820992, 1418569600, 6311953152, 28084886272, 124963582464, 556023840256, 2474023050240, 11008138832896, 48980603529216, 217938687588352

Offset: 0

Johannes W. Meijer, Jul 28 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 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.

The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the corner squares to A179600 and for the side squares to A123347.

Index entries for linear recurrences with constant coefficients, signature (2,10,4).

Cf. A041006, A041007, A123347, A179596, A179597 (central square), A179600.

with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [1,1,1,0,0,0,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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);

G.f.: ( -1-4*x ) / ( (2*x+1)*(2*x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=6 and a(2)=22.
a(n) = (-2/5)*(-1/2)^(-n) + ((2+3*A)*A^(-n-1) + (2+3*B)*B^(-n-1))/10 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1)) for n => 1.

A259594 Denominators of the other-side convergents to sqrt(6).

Original entry on oeis.org

1, 3, 11, 29, 109, 287, 1079, 2841, 10681, 28123, 105731, 278389, 1046629, 2755767, 10360559, 27279281, 102558961, 270037043, 1015229051, 2673091149, 10049731549, 26460874447, 99482086439, 261935653321, 984771132841, 2592895658763, 9748229241971

Offset: 0

Clark Kimberling, Jul 20 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows: p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0. Closeness of P(i)/Q(i) to r is indicated by

|r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = sqrt(6), the first 7 other-side convergents are 3, 7/3, 27/11, 71/29, 267/109, 703/287, 2643/1079. A comparison of convergents with other-side convergents:
i    p(i)/q(i)           P(i)/Q(i)    p(i)*Q(i)-P(i)*q(i)
0    2/1     < sqrt(6) <    3/1               -1
1    5/2     > sqrt(6) >    7/3                1
2    22/9    < sqrt(6) <   27/11              -1
3    49/20   > sqrt(6) >   71/29               1
4    218/89  < sqrt(6) <  267/109             -1
5    485/198 > sqrt(6) >  703/287              1

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A041006, A041007, A259595.

r = Sqrt[6]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t]  (*A259594*)
v = Numerator[t]    (*A259595*)

Vec(-(x+1)*(x^2-2*x-1)/(x^4-10*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
a(n) = 10*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: -(x+1)*(x^2-2*x-1) / (x^4-10*x^2+1). - Colin Barker, Jul 21 2015

A259595 Numerators of the other-side convergents to sqrt(6).

Original entry on oeis.org

3, 7, 27, 71, 267, 703, 2643, 6959, 26163, 68887, 258987, 681911, 2563707, 6750223, 25378083, 66820319, 251217123, 661452967, 2486793147, 6547709351, 24616714347, 64815640543, 243680350323, 641608696079, 2412186788883, 6351271320247, 23878187538507

Offset: 0

Clark Kimberling, Jul 20 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0. Closeness of P(i)/Q(i) to r is indicated by
|r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = sqrt(6), the first 7 other-side convergents are 3, 7/3, 27/11, 71/29, 267/109, 703/287, 2643/1079. A comparison of convergents with other-side convergents:
i    p(i)/q(i)           P(i)/Q(i)    p(i)*Q(i)-P(i)*q(i)
0    2/1     < sqrt(6) <    3/1               -1
1    5/2     > sqrt(6) >    7/3                1
2    22/9    < sqrt(6) <   27/11              -1
3    49/20   > sqrt(6) >   71/29               1
4    218/89  < sqrt(6) <  267/109             -1
5    485/198 > sqrt(6) >  703/287              1

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A041006, A041007, A259594.

r = Sqrt[6]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t]  (* A259594 *)
v = Numerator[t]    (* A259595 *)
LinearRecurrence[{0,10,0,-1},{3,7,27,71},30] (* Harvey P. Dale, Mar 21 2023 *)

Vec((x^3-3*x^2+7*x+3)/(x^4-10*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
a(n) = 10*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: (x^3-3*x^2+7*x+3) / (x^4-10*x^2+1). - Colin Barker, Jul 21 2015

cp's OEIS Frontend

A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 10*x^2 - 4*x^3).

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A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4*x)/(1 - 2*x - 10*x^2 - 4*x^3).

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A259594 Denominators of the other-side convergents to sqrt(6).

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A259595 Numerators of the other-side convergents to sqrt(6).

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Examples

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Mathematica

PARI

Formula

A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 10x^2 - 4*x^3).

A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4x)/(1 - 2x - 10x^2 - 4x^3).