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A258445 Irregular triangle related to Pascal's triangle. 5
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 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A007318 (Pascal's triangle), A164991.
Sequence in context: A226444 A196929 A322494 * A129179 A120621 A201080
KEYWORD
nonn,tabf,easy
AUTHOR
Craig Knecht, May 30 2015
STATUS
approved

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Last modified April 22 10:51 EDT 2024. Contains 371897 sequences. (Running on oeis4.)