A216210 -id:A216210 - cp's OEIS frontend

A216236 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=5, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 0, 0, 5, 10, 10, 4, 0, 0, 5, 15, 20, 14, 0, 0, 0, 0, 20, 35, 34, 14, 0, 0, 0, 0, 20, 55, 69, 48, 0, 0, 0, 0, 0, 0, 75, 124, 117, 48, 0, 0, 0, 0, 0, 0, 75, 199, 241, 165, 0, 0, 0, 0, 0, 0, 0, 0, 274, 440, 406, 165, 0, 0, 0, 0, 0, 0, 0, 0, 274, 714, 846, 571, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

Arithmetic hexagon of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...
1, 2, 3, 4, 5, 5, 0, 0, 0, 0, 0, ...
1, 3, 6, 10, 15, 20, 20, 0, 0, 0, ...
1, 4, 10, 20, 35, 55, 75, 75, 0, 0, 0, ...
0, 4, 14, 34, 69, 124, 199, 274, 274, 0, 0, ...
0, 0, 14, 48, 117, 241, 440, 714, 988, 988, 0, ...
...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris, 1958, Tome 1, p. 89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235.

T(n+3,n) = T(n+2,n) = A094827(n).
T(n+1,n) = A094832(n).
T(n,n) = A094854(n).
T(n,n+1) = A094855(n).
T(n,n+2) = A094833(n+1).
T(n,n+3) = T(n,n+4) = A094828(n).
Sum( T(n-k,k), 0<=k<=n ) = A217733(n). - Philippe Deléham, Mar 22 2013

A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 3, 0, 1, 5, 10, 9, 0, 0, 0, 6, 15, 19, 9, 0, 0, 0, 6, 21, 34, 28, 0, 0, 0, 0, 0, 27, 55, 62, 28, 0, 0, 0, 0, 0, 27, 82, 117, 90, 0, 0, 0, 0, 0, 0, 0, 109, 199, 207, 90, 0, 0, 0, 0, 0, 0, 0, 109, 308, 406, 297, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 24 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, ... row n=0
1, 2, 3, 4, 5, 6, 6, 0, 0, ... row n=1
1, 3, 6, 10, 15, 21, 27, 27, 0, 0, ... row n=2
0, 3, 9, 19, 34, 55, 82, 109, 109, 0, 0, ... row n=3
0, 0, 9, 28, 62, 117, 199, 308, 417, 417, 0, 0, ... row n=4
0, 0, 0, 28, 90, 207, 406, 714, 1131, 1548, 1548, 0, 0, ... row n=5
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 6
1, 3, 6, 10, 15, 21, 27, 27
3, 9, 19, 34, 55, 82, 109, 109
9, 28, 62, 117, 199, 308, 417, 417
28, 90, 207, 406, 714, 1131, 1548, 1548
90, 297, 703, 1417, 2548, 4096, 5644, 5644
297, 1000, 2417, 4965, 9061, 14705, 20349, 20349
1000, 3417, 8382, 17443, 32148, 52497, 72846, 72846
3417, 11799, 29242, 61390, 113887, 186733, 259579, 259579
11799, 41041, 102431, 216318, 403051, 662630, 922209, 922209
...

Cf. Similar sequences: A216201, A216210, A216216, A216218, ...

T(n,n+4) = T(n,n+5) = A094829(n+2).
T(n,n+3) = A094834(n+1).
T(n,n+2) = A094833(n+1).
T(n,n+1) = A094832(n).
T(n,n) = A094831(n).
T(n+1,n) = T(n+2,n) = A094826(n).
sum(T(n-k,k), 0<=k<=n) = A065455(n).

cp's OEIS Frontend

A216236 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=5, T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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