OFFSET
0,5
COMMENTS
This is another "Fibonacci array".
LINKS
Robert Israel, Table of n, a(n) for n = 0..10010 (antidiagonals 0 to 140, flattened)
Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See Table 6.
FORMULA
G.f. as array: 1/((1-x-x*y)*(1-y-y^2)). - Robert Israel, Nov 22 2017
T(n,m) = Sum_{k=0..m} binomial(n+m-k,k). - Vladimir Kruchinin, Jan 14 2024
EXAMPLE
The array begins:
1, 1, 2, 3, 5, 8, 13, 21, 34, ...
1, 2, 3, 5, 8, 13, 21, 34, 55, ...
1, 3, 5, 8, 13, 21, 34, 55, 89, ...
1, 4, 8, 13, 21, 34, 55, 89, 144, ...
1, 5, 12, 21, 34, 55, 89, 144, 233, ...
1, 6, 17, 33, 55, 89, 144, 233, 377, ...
1, 7, 23, 50, 88, 144, 233, 377, 610, ...
1, 8, 30, 73, 138, 232, 377, 610, 987, ...
1, 9, 38, 103, 211, 370, 609, 987, 1597, ...
...
The first few antidiagonals are:
1,
1, 1,
1, 2, 2,
1, 3, 3, 3,
1, 4, 5, 5, 5,
1, 5, 8, 8, 8, 8,
1, 6, 12, 13, 13, 13, 13,
1, 7, 17, 21, 21, 21, 21, 21,
...
MAPLE
A294453:= proc(n, k) option remember;
if n = 0 then combinat:-fibonacci(k+1)
elif k = 0 then 1
else procname(n-1, k-1)+procname(n-1, k)
fi
end proc:
seq(seq(A294453(m-k, k), k=0..m), m=0..10); # Robert Israel, Nov 22 2017
MATHEMATICA
T[0, k_] := Fibonacci[k + 1];
T[_, 0] = 1;
T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k];
Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 16 2020 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Nov 22 2017
EXTENSIONS
Better definition and more terms from Robert Israel, Nov 22 2017
STATUS
approved