cp's OEIS Frontend

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.

%I A355320 #22 May 13 2023 08:27:01
%S A355320 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,
%T A355320 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,
%U A355320 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
%N 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.
%C A355320 See A096608 for the right half of the triangle.
%C A355320 Odd terms form fractal patterns (see illustrations in Links section).
%H A355320 Paolo Xausa, <a href="/A355320/b355320.txt">Table of n, a(n) for n = 0..10010</a> (rows 0..70 of triangle, flattened)
%H A355320 Rémy Sigrist, <a href="/A355320/a355320.png">Representation of the odd terms for n = 0..2^11</a> (only the right half of the triangle is represented)
%H A355320 Rémy Sigrist, <a href="/A355320/a355320_1.png">Representation of the odd terms for n = 0..2^10</a>
%F A355320 T(n, k) = A096608(n, abs(k)).
%F A355320 T(n, 0) = A096609(n).
%F A355320 T(n, 1) = A096610(n).
%F A355320 T(n, 2) = A096611(n).
%F A355320 T(n, n) = A096612(n).
%F A355320 T(n, 2*n) = 1.
%F A355320 T(n, 2*n-1) = T(n, 2*n-2) = 0 for any n > 0.
%F A355320 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).
%F A355320 T(n, -k) = T(n, k).
%F A355320 Sum_{k = -2*n..2*n} T(n, k) = A002605(n+1).
%e A355320 Triangle T(n, k) begins:
%e A355320                                  1
%e A355320                            1  0  0  0  1
%e A355320                      1  0  0  1  2  1  0  0  1
%e A355320                1  0  0  2  3  2  0  2  3  2  0  0  1
%e A355320          1  0  0  3  4  3  1  6  8  6  1  3  4  3  0  0  1
%e A355320    1  0  0  4  5  4  3 12 16 12  6 12 16 12  3  4  5  4  0  0  1
%t A355320 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 *)
%o A355320 (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) }
%Y A355320 Cf. A002605 (row sums), A096608, A096609, A096610, A096611, A096612, A355339.
%K A355320 nonn,tabf,nice,look
%O A355320 0,11
%A A355320 _Rémy Sigrist_, Jun 28 2022