OFFSET
0,4
COMMENTS
T(n,k) is entry k in the Fibonacci-like sequence which starts n, n+1, i.e., T(n,0) = n, T(n,1) = n+1, T(n,k) = T(n,k-1) + T(n,k-2).
Therefore, T(0,k), T(1,k), and T(2,k) are equivalent to A000045(k), A000045(k+2), and A000045(k+3) respectively.
Any two rows T(x,.) and T(y,.) contain infinitely many differing terms for any values of x and y, provided that max(x,y) >= 3.
LINKS
Ely Golden, Table of n, a(n) for n = 0..10000(contains 140 antidiagonals, flattened).
EXAMPLE
T(4,2) = 9 because 9 is element 2 in the Fibonacci-like sequence starting with 4 and 5.
The array begins:
0, 1, 1, 2, 3, 5, 8, ...
1, 2, 3, 5, 8, 13, 21, ...
2, 3, 5, 8, 13, 21, 34, ...
3, 4, 7, 11, 18, 29, 47, ...
4, 5, 9, 14, 23, 37, 60, ...
5, 6, 11, 17, 28, 45, 73, ...
6, 7, 13, 20, 33, 53, 86, ...
PROG
(Python)
def nextAntidiagonal(d):
if(d[0]==0): return [d[1]+1, 0]
return [d[0]-1, d[1]+1]
def fibonacciM(m, n):
f0, f1=0, 1
if(n<0):
for _ in range(-n): f1, f0=f0, f1-f0
else:
for _ in range(n): f0, f1=f1, f0+f1
return f1*m+f0
d=[0, 0]
for i in range(10001):
print(str(i)+" "+str(fibonacciM(d[0], d[1])))
d=nextAntidiagonal(d)
(PARI) T(n, k) = fibonacci(k+1)*n + fibonacci(k);
tabl(nn) = matrix(nn+1, nn+1, i, j, T(i-1, j-1)); \\ Michel Marcus, Sep 27 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ely Golden, Sep 07 2017
STATUS
approved