OFFSET
1,2
COMMENTS
Conjecture: Let F(i) be the i-th Fibonacci number. Each number of T(n, k), k = 1, 2, 3 is the difference between two Fibonacci numbers F(i) - F(j) for some i, j, where F(i) is the smallest Fibonacci number greater than T(n, k). The case T(n, 1) is trivial. Examples: 10 = 13 - 3, 29 = 34 - 5, 20 = 21 - 1, 42 = 55 - 13, 84 = 89 - 5, ...
We observe interesting properties:
T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
T(2n,2) = A033887(n) = 3, 13, 55, ... (Fibonacci(3n+1), and T(2n+1,2) = A048876(n) = 7, 29, 123, ... (Generalized Pell equation with second term of 7) where n = 1, 2, ...
T(3n,3) = 10, 84, 754, 6388,... If n = 2m - 1, T(6m - 3, 3) = F(9m - 2) - F(9m - 5) and if n = 2m, T(6m, 3) = F(9m + 2) - F(9m - 4).
T(3n+1,3) = 20, 178, 1508, 13530, ... If n = 2m - 1, T(6m - 2, 3) = F(9m - 1) - F(9m - 7) and if n = 2m, T(6m+1, 3) = F(9m + 4) - F(9m + 1).
T(3n+2,3) = 42, 356, 3194, 27060, ... If n = 2m - 1, T(6m - 1, 3) = F(9m + 1) - F(9m - 2) and if n = 2m, T(6m + 2, 3) = F(9m + 5) - F(9m - 1).
Other property:
T(2m, 1) + T(2m, 2) = T(2m +1, 1) with T(2m, 1)= F(3m), T(2m, 2) = F(3m + 1) and T(2m + 1, 1) = F(3m + 2).
T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).
EXAMPLE
The start of the sequence as a triangular array T(n, k) read by rows:
1;
2, 3;
5, 7, 10;
8, 13, 20, 30;
21, 29, 42, 62, 92;
34, 55, 84, 126, 188, 280;
...
MAPLE
with(combinat, fibonacci):
lst:={1}:lst2:=lst:
for n from 2 to 15 do :
lst1:={}:ii:=0:
for j from 1 to 1000 while(ii=0) do:
i:=fibonacci(j):
if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
then
lst1:=lst1 union {i}:ii:=1:
else
fi:
od:
for k from 1 to n-1 do:
lst1:=lst1 union {lst1[k]+lst[k]}:
od:
lst:=lst1:lst2:=lst2 union lst:
print(lst1):
od:
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel Lagneau, Jan 02 2019
STATUS
approved