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1, 0, 0, 0, 0, 0, ...
0, 1, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, ...
0, 0, 2, 1, 0, 0, ...
0, 0, 1, 3, 1, 0, ...
0, 0, 0, 3, 4, 1, ...
...
...
... as shown on page 107 of "A Primer for the Fibonacci Numbers".
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
1, 3, 5, 7, 9, ...
1, 4, 7, 10, 13, ...
1, 5, 9, 13, 17, ...
...
...
Leftmost column = Fibonacci numbers, next column (1, 2, 5, 10, 20, ...) = Fibonacci numbers convolved with themselves.
1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, ...
2, 5, 8, 11, 14, ...
3, 10, 17, 24, 31, ...
5, 20, 35, 50, 65, ...
...
...; from which we extract antidiagonals, read by rows, become triangle A106196:
1;
1, 1;
2, 2, 1;
3, 5, 3, 1;
5, 10, 8, 4, 1;
8, 20, 17, 11, 5, 1;
13, 38, 35, 24, 14, 6, 1;
21, 71, 68, 50, 31, 17, 7, 1;
...
...
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Let P = an array with columns comprised composed of Pascal's Triangle rows, offset, spaces filled in with zeros; A = an array composed of arithmetic sequences.(n, k). Perform P * A and extract antidiagonals which become the rows of A106196.
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_Gary W. Adamson (qntmpkt(AT)yahoo.com), _, Apr 24 2005
Triangle read by rows, generated from Pascal's triangle.
...; from which we extract antidiagonals, read by rows, become triangle A106196:
nonn,tabl,new
Let P = an array with columns comprised of Pascal's Triangle rows, offset, spaces filled in with zeros; A = an array composed of arithmetic sequences.(n, k). Perform P * A and extract antidiagonals which become the rows of A106196.
nonn,tabl,new
Triangle by rows, generated from Pascal's triangle.
1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 10, 8, 4, 1, 8, 20, 17, 11, 5, 1, 13, 38, 35, 24, 14, 6, 1, 21, 71, 68, 50, 31, 17, 7, 1
0,4
The array P =
1, 0, 0, 0, 0, 0,...
0, 1, 0, 0, 0, 0,...
0, 1, 1, 0, 0, 0,...
0, 0, 2, 1, 0, 0,...
0, 0, 1, 3, 1, 0,...
0, 0, 0, 3, 4, 1,...
...
...as shown on page 107 of "A Primer for the Fibonacci Numbers".
The array A is composed of arithmetic sequences, as a matrix.
1, 1, 1, 1, 1,...
1, 2, 3, 4, 5,...
1, 3, 5, 7, 9,...
1, 4, 7, 10, 13...
1, 5, 9, 13, 17...
...
Leftmost column = Fibonacci numbers, next column (1, 2, 5, 10, 20...) = Fibonacci numbers convolved with themselves.
V. E. Hoggatt, Jr., editor; "A Primer for the Fibonacci Numbers", 1963, p. 107.
Let P = an array with columns comprised of Pascal's Triangle rows, offset, spaces filled in with zeros; A = an array composed of arithmetic sequences.(n,k). Perform P * A and extract antidiagonals which become the rows of A106196.
The operation P * A generates the array:
1, 1, 1, 1, 1,...
1, 2, 3, 4, 5...
2, 5, 8, 11, 14...
3, 10, 17, 24, 31...
5, 20, 35, 50, 65...
...; from which we extract antidiagonals, by rows, become triangle A106196:
1;
1, 1;
2, 2, 1;
3, 5, 3, 1;
5, 10, 8, 4, 1;
8, 20, 17, 11, 5, 1;
13, 38, 35, 24, 14, 6, 1;
21, 71, 68, 50, 31, 17, 7, 1;
...
nonn,tabl
Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 24 2005
approved