OFFSET
1,5
COMMENTS
LINKS
G. C. Greubel, Antidiagonals n = 1..50, flattened
H. Belbachir and A. Belkhir, Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
A. Dil and I. Mezo, A symmetric algorithm for hyperharmonic and Fibonacci numbers, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Fibonacci numbers.
Piero Filipponi, Incomplete Fibonacci and Lucas numbers, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 1 that contains the incomplete Fibonacci numbers.
Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002), 328-338; see Example 4 (appears as a triangular array).
A. Pintér and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.
FORMULA
G.f.: x*y*(1-x*y)/((x*y+x-1)*(x*y+y-1)). - Mark van Hoeij, Nov 09 2011
From Petros Hadjicostas, Sep 02 2019: (Start)
Following Dil and Mezo (2008), define the incomplete Fibonacci numbers by F(n,k) = Sum_{s = 0..k} binomial(n-1-s, s) for n >= 1 and 0 <= k <= floor((n-1)/2).
Then T(i, j) = F(i+j-1, min(i-1, j-1)) for i,j >= 1.
(End)
EXAMPLE
From Clark Kimberling, Jun 20 2011: (Start)
Northwest corner begins at (i,j) = (1,1):
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, 8, ...
1, 3, 5, 8, 12, 17, 23, 30, ...
1, 4, 8, 13, 21, 33, 50, 73, ...
1, 5, 12, 21, 34, 55, 88, 138, ...
1, 6, 17, 33, 55, 89, 144, 232, ...
1, 7, 23, 50, 88, 144, 233, 377, ...
(End)
Antidiagonal triangle begins as:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 4, 5, 4, 1;
1, 5, 8, 8, 5, 1;
1, 6, 12, 13, 12, 6, 1;
1, 7, 17, 21, 21, 17, 7, 1;
1, 8, 23, 33, 34, 33, 23, 8, 1;
1, 9, 30, 50, 55, 55, 50, 30, 9, 1;
MAPLE
G := x*y*(1-x*y)/((x*y+x-1)*(x*y+y-1)); G := convert(series(G, x=0, 11), polynom):
for i from 1 to 10 do series(coeff(G, x, i), y=0, 11) od; # Mark van Hoeij, Nov 09 2011
# second Maple program:
G:= x*y*(1-x*y)/((x*y+x-1)*(x*y+y-1)):
T:= (i, j)-> coeff(series(coeff(series(G, y, j+1), y, j), x, i+1), x, i):
seq(seq(T(i, 1+d-i), i=1..d), d=1..12); # Alois P. Heinz, Sep 02 2019
# third Maple program:
T:= proc(i, j) option remember; `if`(i=1 or j=1, 1,
max(T(i-1, j) + T(i-1, j-1), T(i-1, j-1) + T(i, j-1)))
end:
seq(seq(T(i, 1+d-i), i=1..d), d=1..12); # Alois P. Heinz, Sep 02 2019
MATHEMATICA
f[i_, 0]:= 1; f[0, i_]:= 1
f[i_, j_]:= f[i, j]= Max[f[i-1, j] +f[i-1, j-1], f[i-1, j-1] +f[i, j-1]];
T[i_, j_]:= f[i-j, j-1];
TableForm[Table[f[i, j], {i, 0, 7}, {j, 0, 7}]]
Table[T[i, j], {i, 10}, {j, i}]//Flatten (* modified by G. C. Greubel, Apr 05 2022 *)
PROG
(Magma)
function t(n, k)
if k eq 0 or n eq 0 then return 1;
else return Max(t(n-1, k-1) + t(n-1, k), t(n-1, k-1) + t(n, k-1));
end if; return t;
end function;
T:= func< n, k | t(n-k, k-1) >;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 05 2022
(SageMath)
def t(n, k):
if (k==0 or n==0): return 1
else: return max(t(n-1, k-1) + t(n-1, k), t(n-1, k-1) + t(n, k-1))
def A038792(n, k): return t(n-k, k-1)
flatten([[A038792(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 05 2022
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, May 02 2000
EXTENSIONS
New description from Benoit Cloitre, Aug 05 2003
Updated from pre-2003 triangular format to present rectangular, from Clark Kimberling, Jun 20 2011
STATUS
approved