for all m, n. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.
A strong divisibility sequence is an integer sequence such that for all positive integers m, n,
Every strong divisibility sequence is a divisibility sequence: if then . Then by the strong divisibility property, and therefore .
- Any constant sequence is a strong divisibility sequence.
- Every sequence of the form , for some nonzero integer k, is a divisibility sequence.
- Every sequence of the form for integers is a divisibility sequence.
- The Fibonacci numbers Fn form a strong divisibility sequence.
- More generally, any Lucas sequence of the first kind Un(P,Q) is a divisibility sequence. Moreover, it is a strong divisibility sequence when gcd(P,Q)=1.
- Elliptic divisibility sequences are another class of such sequences.
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- Hoggatt, Jr., V. E.; Long, C. T. (1973). "Divisibility properties of generalized Fibonacci polynomials" (PDF). Fibonacci Quarterly: 113.
- Bézivin, J.-P.; Pethö, A.; van der Porten, A. J. (1990). "A full characterization of divisibility sequences". Am. J. Math. 112 (6): 985–1001. JSTOR 2374733.
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