A096587 -id:A096587 - cp's OEIS frontend

A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

1, 0, 0, 1, 2, 1, 0, 0, 1, 0, 2, 3, 2, 0, 0, 1, 8, 6, 1, 3, 4, 3, 0, 0, 1, 6, 12, 16, 12, 3, 4, 5, 4, 0, 0, 1, 44, 33, 18, 21, 27, 20, 6, 5, 6, 5, 0, 0, 1, 60, 76, 95, 72, 40, 34, 41, 30, 10, 6, 7, 6, 0, 0, 1, 256, 210, 154, 155, 177, 135, 75, 52, 58, 42, 15, 7, 8, 7, 0, 0, 1, 460, 520, 581, 480

Offset: 0

Clark Kimberling, Jun 29 2004

			Rows:
  1;
  0, 0, 1;
  2, 1, 0, 0, 1;
  0, 2, 3, 2, 0, 0, 1;
T(3,2) counts these paths:
  (0,0)-(1,-2)-(2,0)-(3,2);
  (0,0)-(1,2)-(2,0)-(3,2);
  (0,0)-(1,2)-(2,4)-(3,2).

Paolo Xausa, Table of n, a(n) for n = 0..9999 (rows 0..99 of triangle, flattened)
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Rémy Sigrist, Representation of the odd terms for n = 0..2^11

Cf. A096587, A096609, A096610, A096611, A096612, A355320.

A096608[rowmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[k<=2n,T[n-1,Abs[k-2]]+T[n-2,Abs[k-1]]+T[n-1,k+2]+T[n-2,k+1],0];Table[T[n,k],{n,0,rowmax},{k,0,2n}]]; A096608[10] (* Generates 11 rows *) (* Paolo Xausa, May 09 2023 *)

row(n) = { my (rr=0, r=1); for (k=1, n, [rr, r]=[r, r*(1+'X^4)+rr*('X^3+'X^5)]); Vec(r)[1+2*n..1+4*n] } \\ Rémy Sigrist, Jun 29 2022

T(0, 0) = 1, T(0, 1) = 0, T(0, 2) = 0; T(1, 0) = 0, T(1, 1) = 0, T(1, 2) = 1.
For n >= 2, T(n, 0) = 2*T(n-2, 1) + 2*T(n-1, 2); T(n, 1) = T(n-2, 0) + T(n-2, 2) + T(n-1, 3) + T(n-1, 1); for 2 <= k <= 2n, T(n, k) = T(n-2, k-1) + T(n-2, k+1) + T(n-1, k-2) + T(n-1, k+2).
T(n, 0) + 2*Sum_{k = 1..2*n} T(n, k) = A002605(k). - Rémy Sigrist, Jun 29 2022

Offset changed to 0 by Rémy Sigrist, Jun 29 2022

A099328 Number of Catalan knight paths from (0,0) to (n,0) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 8, 8, 21, 28, 69, 108, 226, 370, 736, 1280, 2473, 4392, 8281, 14920, 27874, 50706, 94088, 171880, 317693, 582116, 1073853, 1970836, 3630914, 6669730, 12279296, 22568896, 41533777, 76360464, 140493041, 258344528, 475256898

Offset: 1

Clark Kimberling, Oct 12 2004

			a(6) counts 8 paths from (0,0) to (6,0); the final move in 5 of the paths is from the point (5,2) and the final move in the other 3 paths is from (4,1).

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,3,1,1,-1).

Cf. A099329, A099330, A099331.

LinearRecurrence[{1,1,-1,3,1,1,-1},{1,0,1,0,2,2,8},40] (* Harvey P. Dale, Aug 11 2017 *)

Taking A099328 to A099331 as the rows of an array T, the recurrences for these row sequences are given for n>=2 by T(n, 0) = T(n-1, 2) + T(n-2, 1), T(n, 1) = T(n-1, 3) + T(n-2, 0) + T(n-2, 2), T(n, 2) = T(n-1, 0) + T(n-2, 1) + T(n-2, 3), T(n, 3) = T(n-1, 1) + T(n-2, 2), with initial values T(0, 0)=1, T(1, 2)=1.
From Chai Wah Wu, Aug 09 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + 3*a(n-4) + a(n-5) + a(n-6) - a(n-7) for n > 7.
G.f.: x*(1 - x - 2*x^4)/((x^4 - 2*x^3 - 1)*(x^3 + x^2 + x - 1)). (End)
2*a(n) = A001590(n)-(-1)^n*( A052922(n-1)+A052922(n-3)) . - R. J. Mathar, Nov 22 2024

A099331 Number of Catalan knight paths from (0,0) to (n,3) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 3, 12, 16, 40, 56, 122, 197, 408, 695, 1352, 2368, 4512, 8096, 15202, 27529, 51196, 93339, 172852, 316368, 584104, 1071160, 1974458, 3625613, 6677104, 12269359, 22583120, 41513728, 76387712, 140454656, 258398850, 475182353

Offset: 0

Clark Kimberling, Oct 12 2004

nonn

			a(6) counts 3 paths from (0,0) to (6,3); the final move in 1
path is from (4,2) and the final move in the other 2 paths
is from (5,1).

Cf. A099328, A099329, A099330.

Taking A099328 to A099331 as the rows of an array T, the recurrences for these row sequences are given for n>=2 by T(n, 0) = T(n-1, 2) + T(n-2, 1), T(n, 1) = T(n-1, 3) + T(n-2, 0) + T(n-2, 2), T(n, 2) = T(n-1, 0) + T(n-2, 1) + T(n-2, 3), T(n, 3) = T(n-1, 1) + T(n-2, 2), with initial values T(0, 0)=1, T(1, 2)=1.
From Chai Wah Wu, Aug 09 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + 3*a(n-4) + a(n-5) + a(n-6) - a(n-7) for n > 6.
G.f.: x^3*(-x^2 + x - 2)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)

A099329 Number of Catalan knight paths from (0,0) to (n,1) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 7, 10, 26, 38, 79, 127, 261, 452, 877, 1540, 2916, 5244, 9837, 17853, 33159, 60486, 111923, 204974, 378334, 694018, 1278939, 2348795, 4325129, 7948424, 14628953, 26893256, 49482888, 90987448, 167388697, 307825273

Offset: 1

Clark Kimberling, Oct 12 2004

nonn

			a(6) counts 7 paths from (0,0) to (6,1); the final move in 4 of the paths is from the point (5,3), the final move in 1 path is from (4,2) and the final move in the other 3 paths is from (4,0).

Cf. A099328, A099330, A099331.

Taking A099328 to A099331 as the rows of an array T, the recurrences for these row sequences are given for n>=2 by T(n, 0) = T(n-1, 2) + T(n-2, 1), T(n, 1) = T(n-1, 3) + T(n-2, 0) + T(n-2, 2), T(n, 2) = T(n-1, 0) + T(n-2, 1) + T(n-2, 3), T(n, 3) = T(n-1, 1) + T(n-2, 2), with initial values T(0, 0)=1, T(1, 2)=1.
From Chai Wah Wu, Aug 09 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + 3*a(n-4) + a(n-5) + a(n-6) - a(n-7) for n > 7.
G.f.: x^3*(x^3 - x^2 - 1)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)

A099330 Number of Catalan knight paths from (0,0) to (n,2) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

Original entry on oeis.org

0, 1, 0, 1, 1, 5, 6, 14, 18, 43, 70, 147, 243, 475, 828, 1596, 2852, 5365, 9676, 18037, 32853, 60929, 111394, 205770, 377142, 695519, 1276818, 2351975, 4320935, 7954167, 14620472, 26904824, 49467208, 91010153, 167357080, 307868201

Offset: 1

Clark Kimberling, Oct 12 2004

nonn

			a(6) counts 6 paths from (0,0) to (6,2); the final move in 1 path is from the point (4,3), the final move in 3 paths is from (4,1) and the final move in the other 2 paths is from (5,0).

Cf. A099328, A099329, A099331.

Taking A099328 to A099331 as the rows of an array T, the recurrences for these row sequences are given for n>=2 by T(n, 0) = T(n-1, 2) + T(n-2, 1), T(n, 1) = T(n-1, 3) + T(n-2, 0) + T(n-2, 2), T(n, 2) = T(n-1, 0) + T(n-2, 1) + T(n-2, 3), T(n, 3) = T(n-1, 1) + T(n-2, 2), with initial values T(0, 0)=1, T(1, 2)=1.
From Chai Wah Wu, Aug 09 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + 3*a(n-4) + a(n-5) + a(n-6) - a(n-7) for n > 7.
G.f.: -x^2*(x^3 - x + 1)/((-x^4 + 2*x^3 + 1)*(x^3 + x^2 + x - 1)). (End)

A096588 a(n)=number of Catalan knight paths in Quadrant I from (0,0) to points on the vertical line x=n. A Catalan knight moves (2 right and 1 up) or (1 right and 1 down).

Original entry on oeis.org

1, 1, 3, 6, 16, 38, 99, 248, 646, 1659, 4342, 11307, 29740, 78115, 206349, 545156, 1445332, 3834559, 10197168, 27140709, 72357778, 193076677, 515843630, 1379308111, 3691755414, 9888374480, 26507373732, 71103941488, 190859621124

Offset: 0

Clark Kimberling, Jun 28 2004

nonn

			Rows of array T(n,k) in A096587:
  1
  0 0 1
  1 1 0 0 1
  0 1 2 2 0 0 1
  ...
so a(3)=T(3,0)+T(3,1)+...+T(3,6)=6.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). Also arXiv:2206.12087, Jan 2023.

Cf. A096587.

A096588list[nmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[0<=k<=2n,T[n-1,k-2]+T[n-2,k-1]+T[n-1,k+2]+T[n-2,k+1],0];Table[Sum[T[n,k],{k,0,2n}],{n,0,nmax}]];A096588list[50] (* Paolo Xausa, May 22 2023 *)

Row sums of array in A096587.

cp's OEIS Frontend

A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

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A099328 Number of Catalan knight paths from (0,0) to (n,0) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

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A099331 Number of Catalan knight paths from (0,0) to (n,3) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

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A099329 Number of Catalan knight paths from (0,0) to (n,1) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

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A099330 Number of Catalan knight paths from (0,0) to (n,2) in the region between and on the lines y=0 and y=3. (See A096587 for the definition of Catalan knight.).

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A096588 a(n)=number of Catalan knight paths in Quadrant I from (0,0) to points on the vertical line x=n. A Catalan knight moves (2 right and 1 up) or (1 right and 1 down).

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