A258445 Irregular triangle related to Pascal's triangle.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 4, 4, 6, 4, 4, 1, 1, 1, 1, 5, 5, 10, 10, 10, 5, 5, 1, 1, 1, 1, 6, 6, 15, 15, 20, 15, 15, 6, 6, 1, 1, 1, 1, 7, 7, 21, 21, 35, 35, 35, 21, 21, 7, 7, 1, 1, 1, 1, 8, 8, 28, 28, 56, 56, 70, 56, 56, 28, 28, 8, 8, 1, 1, 1, 1, 9, 9, 36, 36, 84, 84, 126, 126, 126, 84, 84, 36, 36, 9, 9, 1, 1

Offset: 1

Author

Craig Knecht, May 30 2015

Keywords

• nonn
• tabf
• easy

Comments

The sequence of row lengths of this irregular triangle T(n, k) is A005408(n-1) = 2*n -1.

This array represents the height of water retention between a collection of cylinders whose height and arrangement are specified by Pascal's triangle.

The row sums for this retention are A164991.

Each term is the minimum of 3 terms of Pascal's triangle: 2 terms below and 1 above when k is odd, and 2 terms above and 1 below when k is even. - Michel Marcus, Jun 11 2015

Examples

			The irregular triangle T(n, k) starts:
n\k 1 2 3 4  5  6  7  8  9 10 11 12 13 14 15 16 17
1:  1
2:  1 1 1
3:  1 1 2 1  1
4:  1 1 3 3  3  1  1
5:  1 1 4 4  6  4  4  1  1
6:  1 1 5 5 10 10 10  5  5  1  1
7:  1 1 6 6 15 15 20 15 15  6  6  1  1
8:  1 1 7 7 21 21 35 35 35 21 21  7  7  1  1
9:  1 1 8 8 28 28 56 56 70 56 56 28 28  8  8  1  1
... Reformatted. - _Wolfdieter Lang_, Jun 26 2015
		

Links

• Miguel Angel Amela, Fractal Antenna
• Miguel Angel Amela, Pascal Wave
• Craig Knecht, Pascal's Neighborhood
• Craig Knecht, Pascal Surface
• Craig Knecht, Pascal Cylinders
• Wikipedia, Water Retention on Mathematical Surfaces

Crossrefs

Cf. A007318 (Pascal's triangle), A164991.

Programs

PARI

tabf(nn) = {for (n=1, nn, for (k=1, 2*n-1, kk = (k+1)\2; if (k%2, v = min(binomial(n-1, kk-1), min(binomial(n, kk-1), binomial(n, kk))), v = min(binomial(n, kk), min(binomial(n-1, kk-1), binomial(n-1, kk)))); print1(v, ", ");); print(););} \\ Michel Marcus, Jun 16 2015

Formula

T(n, 2*m) = Min(P(n-1, m-1), P(n-1, m), P(n, m)) with P(n, k) = A007318(n, k) = binomial(n, k), for m = 1, 2, ..., n-1, and

T(n, 2*m-1) = Min(P(n-1, m-1), P(n, m-1), P(n, m)) for m = 1, 2, ..., n. See the program by Michel Marcus. - Wolfdieter Lang, Jun 27 2015