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

0,11

Comments

See A096608 for the right half of the triangle.

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

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

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).

Example

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

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 *)

Prog

(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) }

Crossrefs

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

Sequence in context: A046080 A170967 A035227 * A328556 A321888 A321750

Adjacent sequences: A355317 A355318 A355319 * A355321 A355322 A355323

Keyword

nonn,tabf,nice,look

Author

Rémy Sigrist, Jun 28 2022

Status

approved