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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).
2
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.
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
KEYWORD
nonn,frac,tabl
AUTHOR
Rémy Sigrist, Aug 26 2023
STATUS
approved