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

1,2

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

Links

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

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)).

Example

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

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}]]

Crossrefs

Cf. A214978, A214985, A214986.

Sequence in context: A209150 A209145 A333650 * A118976 A210235 A138177

Adjacent sequences: A214981 A214982 A214983 * A214985 A214986 A214987

Keyword

nonn,tabl

Author

Clark Kimberling, Oct 28 2012

Status

approved