A106196 Triangle read 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

Offset: 0

Author

Gary W. Adamson, Apr 24 2005

Keywords

• nonn
• tabl

Comments

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.

Examples

			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, 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;
  ...
		

References

• V. E. Hoggatt, Jr., editor; "A Primer for the Fibonacci Numbers", 1963, p. 107.

Crossrefs

Cf. A052996, A007678, A106196.

Formula

Let P = an array with columns 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.