Geodesic
Construction of Šindel sequences
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 3, pp. 373-388

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We found that there is a remarkable relationship between the triangular numbers $T_k$ and the astronomical clock (horologe) of Prague. We introduce Šindel sequences $\{a_i\}\subset \Bbb N$ of natural numbers as those periodic sequences with period $p$ that satisfy the following condition: for any $k\in\Bbb N$ there exists $n\in\Bbb N$ such that $T_k=a_1+\cdots+a_n$. We shall see that this condition guarantees a functioning of the bellworks, which is controlled by the horologe. We give a necessary and sufficient condition for a periodic sequence to be a Šindel sequence. We also present an algorithm which produces the so-called primitive Šindel sequence, which is uniquely determined for a given $s=a_1+\cdots+a_p$.
Classification : 01A40, 11A07, 11A51, 11B83
Keywords: Jacobi symbol; quadratic nonresidue; clock sequence; primitive Šindel sequences; Chinese remainder theorem; Dirichlet's theorem 
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     author = {K\v{r}{\'\i}\v{z}ek, Michal and \v{S}olcov\'a, Alena and Somer, Lawrence},
     title = {Construction of {\v{S}indel} sequences},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {373--388},
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Křížek, Michal; Šolcová, Alena; Somer, Lawrence. Construction of Šindel sequences. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 3, pp. 373-388. http://geodesic.mathdoc.fr/item/CMUC_2007__48_3_a0/