A335050 Array read by descending antidiagonals, T(n,k) is the number of nodes in the pill tree with initial conditions (n,k), for n and k >= 0.
1, 2, 3, 3, 7, 8, 4, 12, 21, 22, 5, 18, 40, 63, 64, 6, 25, 66, 130, 195, 196, 7, 33, 100, 231, 427, 624, 625, 8, 42, 143, 375, 803, 1428, 2054, 2055, 9, 52, 196, 572, 1376, 2805, 4860, 6916, 6917, 10, 63, 260, 833, 2210, 5016, 9877, 16794, 23712, 23713
Offset: 0
Examples
The array begins:
1 2 3 4 5 6 ...
3 7 12 18 25 33 ...
8 21 40 66 100 143 ...
22 63 130 231 375 572 ...
64 195 427 803 1376 2210 ...
196 624 1428 2805 5016 8398 ...
...
Links
- Margaret Bayer and Keith Brandt, The Pill Problem, Lattice Paths and Catalan Numbers, preprint, Mathematics Magazine, Vol. 87, No. 5 (December 2014), pp. 388-394.
- Keith Brandt and Kaleb Waite, Using recursion to solve the pill problem, Journal of Computing Sciences in Colleges, Volume 24, Issue 5, May 2009.
- Charlotte A. C. Brennan and Helmut Prodinger, The pills problem revisited, preprint, Quaest. Math., 26(4):427-439, 2003.
- Donald E. Knuth, John McCarthy, Walter Stromquist, Daniel H. Wagner, and Tim Hesterberg, Problem E3429. Big pills and little pills, The American Mathematical Monthly, 99(7):684, 1992.
Crossrefs
Programs
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PARI
T(n,k) = sum(j=0, n, binomial(2*j+k+2, j+1) - binomial(2*j+k+2, j));
Formula
T(0,k) = k+1; T(n,0) = 1 + T(n-1,1); T(n,k) = 1 + T(n-1,k+1) + T(n,k-1) for n and k > 0.
T(n,k) = Sum_{j=0..n} (binomial(2*j+k+2, j+1) - binomial(2*j+k+2, j)).
Comments