This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Table begins: n\k| 1 2 3 4 5 6 7 8 ---+---------------------------------------------------------- 1| 1 2 10 63 454 3539 29008 246255 2| 1 5 33 240 1871 15314 129825 1129967 3| 2 13 98 777 6420 54758 478662 4266102 4| 4 32 269 2295 19970 176971 1593093 14532881 5| 8 76 702 6393 58342 536080 4965056 46345046 6| 16 176 1768 17088 163041 1550809 14765863 140982374 7| 32 400 4336 44280 440602 4332221 42373370 413689403 8| 64 896 10416 111984 1159580 11771312 118190333 1179448443 For example, the T(2,2) = 5 sequences of permissible queen's moves from (1,1) to (2,2) are: (1,1) -> (1,2) -> (2,2), (1,1) -> (2,1) -> (1,2) -> (2,2), (1,1) -> (2,1) -> (2,2), (1,1) -> (2,1) -> (3,1) -> (2,2), and (1,1) -> (3,1) -> (2,2).
The array begins 1 2 7 30 ... 1 4 20 ... 1 8 ... 1 ...
There are four rook paths with move length capped by the number of the rank or file it is moving along, from (1,1) to (3,2):
(1,1)->(2,1)->(3,1)->(3,2);
(1,1)->(2,1)->(2,2)->(3,2);
(1,1)->(1,2)->(2,2)->(3,2);
(1,1)->(1,2)->(3,2).
So T(3,2) = 4.
An initial portion of the full array:
n= 1 2 3 4 5 6 7 8 9 ...
-----------------------------------------
k=1: 1 1 1 1 1 1 1 1 1 ...
k=2: 1 2 4 7 12 20 33 54 88 ...
k=3: 1 4 10 23 50 104 211 420 824 ...
k=4: 1 7 23 62 156 373 859 1925 4226 ...
k=5: 1 12 50 156 438 1155 2915 7114 16917 ...
k=6: 1 20 104 373 1155 3306 8978 23450 59422 ...
....
n = 1; k = 1;
T = [[],[0]] # Dummy 0th entry, and dummy [1][0]th entry.
T[n].append(1) # set T[1][1] to 1
print(f"T(1,1) = {T[n][k]}")
for m in range(64):
if n == 1:
n = k + 1; k = 1;
T.append([0]); # initialize T[n], with dummy 0th entry.
else:
n -= 1; k += 1;
T[n].append(sum(T[i][k] for i in range(max(1,n-k),n))
+ sum(T[n][j] for j in range(max(1,k-n),k)))
print(f"T({n},{k}) = {T[n][k]}")
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