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==Compositions of natural numbers over arithmetic progressions==
Denote by
Each m-sequence with terms in set <math>A_{s}^{p}\,</math> .If sum of terms is equal at k is called composition of natural number k in m parts
Denote by :<math>C_{m}(k,A_{s}^{p})=\{x:x=si+p,i\in N,p\in I_{s}\}\,</math>
the set
Each m-sequence of natural numbers that fulfill the conditions.
:<math>c_{0}+c_{1}+...c_{m-1}=k\,</math>, <math>c_{i}\in A_{s}^{p}\,</math>
is called composition of natural number k in m parts over <math>A_{s}^{p}\,</math>
<br>Denote by <math>C_{m}(k,A_{s}^{p})\,</math> the set of compositions of natural number k in m parts over <math>A_{s}^{p}\,</math>