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.
[1] Another way of describing such a sequence is that it can be partitioned into contiguous subsequences whose sums are the consecutive integers:[2] 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.
The Šindel sequences are named after Jan Šindel, a Czech scientist in the 14th and 15th centuries whose calculations were used in the design of the Prague clock.
[3][4] The definition and name of these sequences were given by Michal Křížek, Alena Šolcová, and Lawrence Somer, in their work analyzing the mathematics of the Prague clock.
need to be checked, to determine whether it is a Šindel sequence.
If all of these triangular numbers are partial sums of the sequence, then all larger triangular numbers will be as well.
, a larger set of triangular numbers needs to be checked, up to
[7] In the Prague clock, an auxiliary gear with slots spaced at intervals of 1, 2, 3, 4, 3, and 2 units (repeating in the example Šindel sequence in each of its rotations) is synchronized with and regulates the motion of another larger gear whose slots are spaced at intervals of 1, 2, 3, 4, 5, ..., 24 units, revolving once a day with its spacing controlling the number of chimes on each hour.
of the period of the Šindel sequence must evenly divide
, the sum of spacing intervals of the large gear.
[3] For this reason it is of interest to find Šindel sequences with a given period sum
In connection with this problem, a primitive Šindel sequence, is a Šindel sequence no two of whose numbers can be replaced by their sum, forming a shorter Šindel sequence.
there exists a unique primitive Šindel sequence having period sum equal to
is a power of two, then the trivial Šindel sequence with period
, the unique primitive Šindel sequence with period sum