A214984 Array: T(m,n) = (F(m) + F(2*m) + ... + F(n*m))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

1, 2, 1, 4, 4, 1, 7, 12, 5, 1, 12, 33, 22, 8, 1, 20, 88, 94, 56, 12, 1, 33, 232, 399, 385, 134, 19, 1, 54, 609, 1691, 2640, 1487, 342, 30, 1, 88, 1596, 7164, 18096, 16492, 6138, 872, 48, 1, 143, 4180, 30348, 124033, 182900, 110143, 25319, 2256, 77, 1

Offset: 1

Author

Clark Kimberling, Oct 28 2012

Keywords

• nonn
• tabl

Comments

col 1: A001612 (except for initial term)

row 1: A000071

row 2: A027941

row 3: A049652

row 4: A092521

row 6: A049664

row 8: A156093 without minus signs

Examples

			Northwest corner:
1...2....4.....7......12......20
1...4....12....33.....88......232
1...5....22....94.....399.....1691
1...8....56....385....2640....18096
1...12...134...1487...16492...182900
		

Links

• Clark Kimberling, Antidiagonals n = 1..60, flattened

Crossrefs

Cf. A214978, A214985, A214986.

Programs

Mathematica

F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
t[m_, n_] := (1/F[m])*Sum[F[m*k], {k, 1, n}]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]

Formula

For odd-numbered rows (m odd):

T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).

For even-numbered rows (m even):

T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-2)).