OFFSET
1,7
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
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
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
EXAMPLE
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
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Craig Knecht, May 30 2015
STATUS
approved