A365212 Triangle T(n, k), n >= 0, k = 0..n, read by rows: let's consider a triangle of initially empty glasses of equal volume, G(n, k), n >= 0, k = 0..n; when water is poured into one of the glasses, say G(n, k), it flows into that glass until it's full, and then the excess overflows equally into G(n+1, k) and G(n+1, k+1); let V(n, k) be the minimum volume of water to be poured into G(0, 0) so as to fill G(n, k) completely; T(n, k) is the numerator of V(n, k) / V(0, 0).

1, 3, 3, 7, 5, 7, 15, 25, 25, 15, 31, 41, 11, 41, 31, 63, 22, 77, 77, 22, 63, 127, 183, 109, 93, 109, 183, 127, 255, 311, 31, 403, 403, 31, 311, 255, 511, 105, 226, 779, 467, 779, 226, 105, 511, 1023, 1269, 1853, 2253, 100, 100, 2253, 1853, 1269, 1023

Offset

0,2

Comments

See A365213 for the denominators.

Links

Table of n, a(n) for n=0..54.

Rémy Sigrist, PARI program

Formula

T(n, k) = T(n, n - k).

T(n, 0) = 2^(n+1) - 1.

Example

Triangle T(n, k) begins:
                                  1
                               3     3
                            7     5     7
                        15    25    25    15
                     31    41    11    41    31
                  63    22    77    77    22    63
               127   183   109   93    109   183   127
            255   311   31    403   403   31    311   255
         511   105   226   779   467   779   226   105   511
.
Triangle V(n, k) / V(0, 0) begins:
                                   1
                               3       3
                           7       5       7
                      15     25/3    25/3     15
                  31     41/3     11     41/3     31
              63      22     77/5    77/5     22      63
          127    183/5   109/5   93/5    109/5   183/5    127
      255    311/5    31    403/17  403/17    31     311/5    255
  511     105    226/5  779/25  467/17  779/25   226/5    105     511

Prog

(PARI) See Links section.

Crossrefs

Cf. A365213 (denominators).

Sequence in context: A218388 A308340 A161427 * A098688 A129266 A129527

Adjacent sequences: A365209 A365210 A365211 * A365213 A365214 A365215

Keyword

nonn,frac,tabl

Author

Rémy Sigrist, Aug 26 2023

Status

approved