Šindel sequence
In additive combinatorics, a Šindel sequence is a periodic sequence of integers with the property that its partial sums include all of the triangular numbers. For instance, the sequence that begins 1, 2, 3, 4, 3, 2 is a Šindel sequence, with the triangular partial sums 1 = 1 , 3 = 1 + 2 , 6 = 1 + 2 + 3 , 10 = 1 + 2 + 3 + 4 , 15 = 1 + 2 + 3 + 4 + 3 + 2 , 21 = 1 + 2 + 3 + 4 + 3 + 2 + 1 + 2 + 3 , 28 = 1 + 2 + 3 + 4 + 3 + 2 + 1 + 2 + 3 + 4 + 3 , ⋮ {\displaystyle {\begin{aligned}1&=1,\\[2pt]3&=1+2,\\[2pt]6&=1+2+3,\\[2pt]10&=1+2+3+4,\\[2pt]15&=1+2+3+4+3+2,\\[2pt]21&=1+2+3+4+3+2+1+2+3,\\[2pt]28&=1+2+3+4+3+2+1+2+3+4+3,\\&\ \ \vdots \end{aligned}}} Another way of describing such a sequence is that it can be partitioned into contiguous subsequences whose sums are the consecutive integers: 1 = 1 , 2 = 2 , 3 = 3 , 4 = 4 , 5 = 3 + 2 , 6 = 1 + 2 + 3 , 7 = 4 + 3 , 8 = 2 + 1 + 2 + 3 , 9 = 4 + 3 + 2 , ⋮ {\displaystyle {\begin{aligned}1&=1,\\[2pt]2&=2,\\[2pt]3&=3,\\[2pt]4&=4,\\[2pt]5&=3+2,\\[2pt]6&=1+2+3,\\[2pt]7&=4+3,\\[2pt]8&=2+1+2+3,\\[2pt]9&=4+3+2,\\&\ \ \vdots \end{aligned}}} This particular example is used in the gearing of the Prague astronomical clock, as part of a mechanism for chiming the clock's bells the correct number of times at each hour.