OFFSET
1,2
COMMENTS
LINKS
Clark Kimberling, Antidiagonals n=1..60, flattened
FORMULA
T(n,k) = 3*T(n,k-1)-2*T(n,k-2)-T(n,k-3)+T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*(2*n - 1 + 2*x - (2*n - 3)*x^2) and g(x) = (1 - x - x^2)*(1 - x )^2.
T(n,k) = 2*n*Fibonacci(k+3) + Lucas(k+3) - 4*(k+n+1). - Ehren Metcalfe, Jul 08 2019
EXAMPLE
Northwest corner (the array is read by falling antidiagonals):
1....5....14...31....61....112
3....11...26...53....99....176
5....17...38...75....137...240
7....23...50...97....175...304
9....29...62...119...213...368
11...35...74...141...251...432
MATHEMATICA
b[n_] := Fibonacci[n + 1]; c[n_] := 2 n - 1;
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}] (* A213774 *)
Table[t[n, n], {n, 1, 40}] (* A213775 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213776 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jun 21 2012
STATUS
approved