A365213 - OEIS

A365213

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 denominator of V(n, k) / V(0, 0).

%I #5 Aug 30 2023 09:20:09

%S 1,1,1,1,1,1,1,3,3,1,1,3,1,3,1,1,1,5,5,1,1,1,5,5,5,5,5,1,1,5,1,17,17,

%T 1,5,1,1,1,5,25,17,25,5,1,1,1,7,27,55,3,3,55,27,7,1,1,2,11,75,9,25,9,

%U 75,11,2,1,1,9,35,55,50,215,215,50,55,35,9,1

%N 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 denominator of V(n, k) / V(0, 0).

%C See A365212 for the numerators.

%H Rémy Sigrist, <a href="/A365213/a365213.gp.txt">PARI program</a>

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

%F T(n, 0) = 1.

%e Triangle T(n, k) begins:

%e 1

%e 1 1

%e 1 1 1

%e 1 3 3 1

%e 1 3 1 3 1

%e 1 1 5 5 1 1

%e 1 5 5 5 5 5 1

%e 1 5 1 17 17 1 5 1

%e 1 1 5 25 17 25 5 1 1

%e .

%e Triangle V(n, k) / V(0, 0) begins:

%e 1

%e 3 3

%e 7 5 7

%e 15 25/3 25/3 15

%e 31 41/3 11 41/3 31

%e 63 22 77/5 77/5 22 63

%e 127 183/5 109/5 93/5 109/5 183/5 127

%e 255 311/5 31 403/17 403/17 31 311/5 255

%e 511 105 226/5 779/25 467/17 779/25 226/5 105 511

%o (PARI) See Links section.

%Y Cf. A365212 (numerators).

%K nonn,frac,tabl

%O 0,8

%A _Rémy Sigrist_, Aug 26 2023