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

0,4

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.

References

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

Links

Table of n, a(n) for n=0..35.

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.

Example

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;
  ...

Crossrefs

Cf. A052996, A007678, A106196.

Sequence in context: A081572 A292630 A144287 * A037027 A182810 A139375

Adjacent sequences: A106193 A106194 A106195 * A106197 A106198 A106199

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 24 2005

Status

approved