A096612 -id:A096612 - 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

A096609 a(n)=number of Catalan knight paths in right half-plane from (0,0) to (n,0).

Original entry on oeis.org

1, 0, 2, 0, 8, 6, 44, 60, 256, 460, 1582, 3360, 10324, 24150, 69652, 172200, 479728, 1225308, 3350972, 8727312, 23638174, 62295420, 167923252, 445808220, 1199180360, 3198756132, 8598547622, 23009417080, 61856381708, 165897510350

Offset: 0

Clark Kimberling, Jun 29 2004

nonn

Column 0 of the array in A096608.

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A096608, A096610, A096611, A096612.

A096609[nmax_]:=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,0],{n,0,nmax}]];A096609[50] (* Paolo Xausa, May 09 2023 *)

A096610 a(n)=number of Catalan knight paths in right half-plane from (0,0) to (n,1).

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 33, 76, 210, 520, 1410, 3590, 9618, 24920, 66521, 174412, 465264, 1229184, 3280458, 8710668, 23271116, 62014876, 165888184, 443255164, 1187255082, 3178905184, 8525398102, 22864654046, 61391533270, 164872981784

Offset: 0

Clark Kimberling, Jun 29 2004

nonn

Column 1 of the array in A096608.

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A096608, A096609, A096611, A096612.

A096610[nmax_]:=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,1],{n,0,nmax}]];A096610[50] (* Paolo Xausa, May 09 2023 *)

A096611 a(n)=number of Catalan knight paths in right half-plane from (0,0) to (n,2).

Original entry on oeis.org

0, 1, 0, 3, 1, 16, 18, 95, 154, 581, 1160, 3752, 8485, 25208, 61180, 173343, 438242, 1210222, 3134472, 8538629, 22437042, 60690510, 160889234, 433701996, 1156122902, 3112018729, 8325803356, 22402792752, 60084101129, 161701735578

Offset: 0

Clark Kimberling, Jun 29 2004

nonn

Column 2 of the array in A096608.

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A096608, A096609, A096610, A096612.

A096611[nmax_]:=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,2],{n,0,nmax}]];A096611[50] (* Paolo Xausa, May 09 2023 *)

A355320 Irregular triangle T(n, k), n >= 0, -2n <= k <= 2n, read by rows; T(0, 0) = 1; for n > 0, T(n, k) is the sum of all terms in previous rows at one knight's move away.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 2, 3, 2, 0, 2, 3, 2, 0, 0, 1, 1, 0, 0, 3, 4, 3, 1, 6, 8, 6, 1, 3, 4, 3, 0, 0, 1, 1, 0, 0, 4, 5, 4, 3, 12, 16, 12, 6, 12, 16, 12, 3, 4, 5, 4, 0, 0, 1, 1, 0, 0, 5, 6, 5, 6, 20, 27, 21, 18, 33, 44, 33, 18, 21, 27, 20, 6, 5, 6, 5, 0, 0, 1

Offset: 0

Rémy Sigrist, Jun 28 2022

See A096608 for the right half of the triangle.

Odd terms form fractal patterns (see illustrations in Links section).

			Triangle T(n, k) begins:
                                 1
                           1  0  0  0  1
                     1  0  0  1  2  1  0  0  1
               1  0  0  2  3  2  0  2  3  2  0  0  1
         1  0  0  3  4  3  1  6  8  6  1  3  4  3  0  0  1
   1  0  0  4  5  4  3 12 16 12  6 12 16 12  3  4  5  4  0  0  1

Paolo Xausa, Table of n, a(n) for n = 0..10010 (rows 0..70 of triangle, flattened)
Rémy Sigrist, Representation of the odd terms for n = 0..2^11 (only the right half of the triangle is represented)
Rémy Sigrist, Representation of the odd terms for n = 0..2^10

Cf. A002605 (row sums), A096608, A096609, A096610, A096611, A096612, A355339.

A355320[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,Abs[k]],{n,0,rowmax},{k,-2n,2n}]]; A355320[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) }

T(n, k) = A096608(n, abs(k)).
T(n, 0) = A096609(n).
T(n, 1) = A096610(n).
T(n, 2) = A096611(n).
T(n, n) = A096612(n).
T(n, 2*n) = 1.
T(n, 2*n-1) = T(n, 2*n-2) = 0 for any n > 0.
T(n, k) = T'(n-1, k-2) + T'(n-1, k+2) + T'(n-2, k-1) + T'(n-2, k+1) for n > 0 (where T' extends T with 0's outside its domain of definition).
T(n, -k) = T(n, k).
Sum_{k = -2*n..2*n} T(n, k) = A002605(n+1).

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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A096609 a(n)=number of Catalan knight paths in right half-plane from (0,0) to (n,0).

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A096610 a(n)=number of Catalan knight paths in right half-plane from (0,0) to (n,1).

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A096611 a(n)=number of Catalan knight paths in right half-plane from (0,0) to (n,2).

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A355320 Irregular triangle T(n, k), n >= 0, -2*n <= k <= 2*n, read by rows; T(0, 0) = 1; for n > 0, T(n, k) is the sum of all terms in previous rows at one knight's move away.

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A355320 Irregular triangle T(n, k), n >= 0, -2n <= k <= 2n, read by rows; T(0, 0) = 1; for n > 0, T(n, k) is the sum of all terms in previous rows at one knight's move away.