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

%I A214984 #12 Jan 09 2024 16:39:40
%S A214984 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,
%T A214984 134,19,1,54,609,1691,2640,1487,342,30,1,88,1596,7164,18096,16492,
%U A214984 6138,872,48,1,143,4180,30348,124033,182900,110143,25319,2256,77,1
%N A214984 Array: T(m,n) = (F(m) + F(2*m) + ... + F(n*m))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).
%C A214984 col 1: A001612 (except for initial term)
%C A214984 row 1: A000071
%C A214984 row 2: A027941
%C A214984 row 3: A049652
%C A214984 row 4: A092521
%C A214984 row 6: A049664
%C A214984 row 8: A156093 without minus signs
%H A214984 Clark Kimberling, <a href="/A214984/b214984.txt">Antidiagonals n = 1..60, flattened</a>
%F A214984 For odd-numbered rows (m odd):
%F A214984 T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
%F A214984 For even-numbered rows (m even):
%F A214984 T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-2)).
%e A214984 Northwest corner:
%e A214984 1...2....4.....7......12......20
%e A214984 1...4....12....33.....88......232
%e A214984 1...5....22....94.....399.....1691
%e A214984 1...8....56....385....2640....18096
%e A214984 1...12...134...1487...16492...182900
%t A214984 F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
%t A214984 t[m_, n_] := (1/F[m])*Sum[F[m*k], {k, 1, n}]
%t A214984 TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
%t A214984 Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]
%Y A214984 Cf. A214978, A214985, A214986.
%K A214984 nonn,tabl
%O A214984 1,2
%A A214984 _Clark Kimberling_, Oct 28 2012