A038792 Rectangular array defined by T(i,1) = T(1,j) = 1 for i >= 1 and j >= 1; T(i,j) = max(T(i-1,j) + T(i-1,j-1), T(i-1,j-1) + T(i,j-1)) for i >= 2, j >= 2, read by antidiagonals.
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, 1, 10, 38, 73, 88, 89, 88, 73, 38, 10, 1, 1, 11, 47, 103, 138, 144, 144, 138, 103, 47, 11, 1
Offset: 1
Examples
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;
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.
Crossrefs
Programs
-
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
-
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 *) -
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
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)
Extensions
New description from Benoit Cloitre, Aug 05 2003
Updated from pre-2003 triangular format to present rectangular, from Clark Kimberling, Jun 20 2011
Comments