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
etc.[1] Another way of describing such a sequence is that it can be partitioned into contiguous subsequences whose sums are the consecutive integers:[2]
| subsequence | 1 | 2 | 3 | 4 | 3, 2 | 1, 2, 3 | 4, 3 | 2, 1, 2, 3 | 4, 3, 2 | ... |
|---|---|---|---|---|---|---|---|---|---|---|
| sum | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... |
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.[5]
If [math]\displaystyle{ s }[/math] denotes the sum of the numbers within a single period of a periodic sequence, and [math]\displaystyle{ s }[/math] is odd, then only the triangular numbers [math]\displaystyle{ \tbinom{i}{2} }[/math] up to [math]\displaystyle{ \tbinom{(s+1)/2}{2} }[/math] 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.[6] For even values of [math]\displaystyle{ s }[/math], a larger set of triangular numbers needs to be checked, up to [math]\displaystyle{ \tbinom{s}{2} }[/math].[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. In order to keep these two gears synchronized, it is important that, for every revolution of the large gear, the small gear also revolves an integer number of times. Mathematically, this means that the sum [math]\displaystyle{ s=15 }[/math] of the period of the Šindel sequence must evenly divide [math]\displaystyle{ \tbinom{25}{2} }[/math], 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 [math]\displaystyle{ s }[/math]. 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. For every [math]\displaystyle{ s }[/math] there exists a unique primitive Šindel sequence having period sum equal to [math]\displaystyle{ s }[/math]. Note however, that this sequence may be formed by repeating a shorter Šindel sequence more than one time.[8]
A sequence that just repeats the number 1, with any period, is a Šindel sequence, and is called the trivial Šindel sequence. If [math]\displaystyle{ s }[/math] is a power of two, then the trivial Šindel sequence with period [math]\displaystyle{ s }[/math] is primitive, and is the unique primitive Šindel sequence with period sum [math]\displaystyle{ s }[/math]. For any other choice of [math]\displaystyle{ s }[/math], the unique primitive Šindel sequence with period sum [math]\displaystyle{ s }[/math] is not trivial.[9]
See also
Notes
- ↑ Křížek, Somer & Šolcová (2021), Equation 10.5, p. 229.
- ↑ Křížek, Somer & Šolcová (2021), Equation 10.6, p. 229.
- ↑ 3.0 3.1 Křížek, Somer & Šolcová (2021), p. 225.
- ↑ Orloj Cog and Orloj Cog: Solution, Problem of the Week, Fall 2021 Week 6, University of Nebraska Omaha Department of Mathematics, retrieved 2021-12-27
- ↑ Křížek, Somer & Šolcová (2006).
- ↑ Křížek, Somer & Šolcová (2021), Theorem 10.1, p. 229.
- ↑ Křížek, Somer & Šolcová (2021), Theorem 10.2, p. 232.
- ↑ Křížek, Somer & Šolcová (2021), Theorem 10.6, p. 237.
- ↑ Křížek, Somer & Šolcová (2021), Theorem 10.7, p. 238.
References
- Chleboun, Jan (2012), "Michal’s roses of Jericho", in Brandts, Jan; Chleboun, J.; Korotov, Sergej et al., Applications of Mathematics 2012, In honor of the 60th birthday of Michal Křížek, Proceedings, Prague, May 2-5, 2012, Prague: Institute of Mathematics AS CR, pp. xxxi–xxxiii, https://dml.cz/dmlcz/702886
- Křížek, Michal; Somer, Lawrence; Šolcová, Alena (May 2006), "Šindel sequences and the Prague horologe", in Chleboun, Jan; Segeth, Karel; Vejchodský, Tomáš, Programs and Algorithms of Numerical Mathematics, Proceedings of Seminar, Prague, Prague: Institute of Mathematics AS CR, pp. 156–164, https://dml.cz/dmlcz/702831
- Křížek, Michal; Somer, Lawrence; Šolcová, Alena (2021), "Chapter 10: The mathematics behind Prague's horologe", From Great Discoveries in Number Theory to Applications, Springer International Publishing, pp. 225–252, doi:10.1007/978-3-030-83899-7_10
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