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.
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
Examples
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
Links
- 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
Programs
-
Mathematica
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 *)
-
PARI
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) }
Formula
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).
Comments