|
|
EXAMPLE
|
Triangle T begins:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
310, 105, 25, 5, 1, 1;
10978, 2702, 480, 68, 8, 1, 1;
868140, 154609, 20657, 2184, 182, 13, 1, 1;
149688297, 19092682, 1906051, 152579, 9562, 483, 21, 1, 1;
57339888914, 5161046609, 378639419, 22799907, 1090125, 41480, 1275, 34, 1, 1; ...
GENERATE T FROM MATRIX POWERS OF T.
Row n of T = row n-1 of T^fibonacci(n) with appended '1'.
Examples.
Row 5 of T is given by row 4 of matrix power T^fibonacci(5) = T^5:
1;
5, 1;
15, 5, 1;
55, 20, 5, 1;
310, 105, 25, 5, 1; <== row 5 of T
3796, 1070, 215, 35, 5, 1; ...
Row 6 of T is given by row 5 of matrix power T^fibonacci(6) = T^8:
1;
8, 1;
36, 8, 1;
164, 44, 8, 1;
978, 268, 52, 8, 1;
10978, 2702, 480, 68, 8, 1; <== row 6 of T
262838, 53648, 8082, 964, 92, 8, 1; ...
ALTERNATE GENERATING METHOD.
To obtain row n, start with a '1' repeated fibonacci(n) times,
and build a table where row k+1 equals the partial sums of row k
but with the last term appearing fibonacci(n-k) times, for k=1..n-1;
listing the final terms in each row forms row n of this triangle.
Example.
To obtain row 5, start with a '1' repeated fibonacci(5)=5 times:
(1,1,1,1,1);
take partial sums, writing the last term fibonacci(4)=3 times:
1,2,3,4, (5,5,5);
take partial sums, writing the last term fibonacci(3)=2 times:
1,3,6,10,15,20, (25,25);
take partial sums, writing the last term fibonacci(2)=1 times:
1,4,10,20,35,55,80, (105);
take partial sums, writing the last term fibonacci(1)=1 times:
1,5,15,35,70,125,205, (310).
Final terms in the above partial sums forms row 5: [310,105,25,5,1];
repeating this process will generate all the rows of this triangle.
|
