%I #48 Dec 02 2018 23:04:26

%S 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,

%T 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,

%U 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

%N Irregular triangle related to Pascal's triangle.

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

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

%C The row sums for this retention are A164991.

%C 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

%H Miguel Angel Amela, <a href="/A258445/a258445_2.png">Fractal Antenna</a>

%H Miguel Angel Amela, <a href="/A258445/a258445_1.png">Pascal Wave</a>

%H Craig Knecht, <a href="/A258445/a258445_2.jpg">Pascal's Neighborhood</a>

%H Craig Knecht, <a href="/A258445/a258445.jpg">Pascal Surface</a>

%H Craig Knecht, <a href="/A258445/a258445_1.jpg">Pascal Cylinders</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water retention on mathematical surfaces">Water Retention on Mathematical Surfaces</a>

%F 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

%F 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

%e The irregular triangle T(n, k) starts:

%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

%e 1: 1

%e 2: 1 1 1

%e 3: 1 1 2 1 1

%e 4: 1 1 3 3 3 1 1

%e 5: 1 1 4 4 6 4 4 1 1

%e 6: 1 1 5 5 10 10 10 5 5 1 1

%e 7: 1 1 6 6 15 15 20 15 15 6 6 1 1

%e 8: 1 1 7 7 21 21 35 35 35 21 21 7 7 1 1

%e 9: 1 1 8 8 28 28 56 56 70 56 56 28 28 8 8 1 1

%e ... Reformatted. - _Wolfdieter Lang_, Jun 26 2015

%o (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

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

%K nonn,tabf,easy

%O 1,7

%A _Craig Knecht_, May 30 2015