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This array aims the study of the divisibility properties of the Fibonacci numbers A352744. The identity F(n, k) = (-1)^k*F(1 - n, -k) from A352744 shows that negative indices do not add to the divisibility properties of F(n, k).

All rows A(n, .) are pure periodic sequences. The length of the periods is given by (1, A270313). For n > 0 the length of the period of row A(n, .) is <= n.

The period length is 1 for n in (1, A023172) and n for n in (1, A074215), as observed by Robert Israel in A270313. In particular, if n is a power of 2 or a prime (A174090), then the period length is n.

The indices of the zero-free rows are in A353280. A zero-free row A(n, .) means that n will not divide F(k, n) whatever value k takes. For that it is sufficient to check that period(A(n, .)) is zero-free.

If period(A(n, .)) = [k | 0 <= k < n] we call n a 'Fibonacci friend'. In other words, in this case F(k, n) mod n = k for 0 <= k < n. A Fibonacci friend does not have to be prime (since 1 is a Fibonacci friend), but if it is prime then it is congruent to {1, 4} mod 5 (A045468), and all such primes are Fibonacci friends.

To say that n is a Fibonacci friend is equivalent to saying that A(n, n) = 0 and that n divides F(n, n). Fibonacci friends are the indices of the zeros in A002752.

Integers n > 0 that divide Sum{k=0..n-1} (F(k, n) mod n) are congruent to {0, 1, 3, 5} mod 6 (A301729).

n is a term if 0 is not a term of the sequence A352747(n, .). Since A352747(n, .) is for all n a pure periodic sequence, it is sufficient to require that 0 is not a term of period(A352747(n, .)). Since the length of the period is <= n, the condition can be checked in a finite number of steps.

The multiples of 5 and 6 (A093509) are a subsequence. The terms not of this form start 56, 91, 112, ..., and are in A353281.

From Gus Wiseman, Feb 12 2019: (Start)

Also the number of ways to split an (n + 1)-cycle into nonempty connected subgraphs with no singletons. For example, the a(1) = 1 through a(5) = 12 partitions are:

{{12}} {{123}} {{1234}} {{12345}} {{123456}}

{{12}{34}} {{12}{345}} {{12}{3456}}

{{14}{23}} {{123}{45}} {{123}{456}}

{{125}{34}} {{1234}{56}}

{{145}{23}} {{1236}{45}}

{{15}{234}} {{1256}{34}}

{{126}{345}}

{{1456}{23}}

{{156}{234}}

{{16}{2345}}

{{12}{34}{56}}

{{16}{23}{45}}

Also the number of non-singleton subsets of {1, ..., (n + 1)} with no cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(1) = 1 through a(5) = 12 subsets are:

{} {} {} {} {}

{1,3} {1,3} {1,3}

{2,4} {1,4} {1,4}

{2,4} {1,5}

{2,5} {2,4}

{3,5} {2,5}

{2,6}

{3,5}

{3,6}

{4,6}

{1,3,5}

{2,4,6}

(End)

The definition declares the Fibonacci numbers for all integers n and k. It gives the classical Fibonacci numbers as F(0, n) = A000045(n). A different enumeration is given in A352744.

The definition declares the Lucas numbers for all integers n and k. It gives the classical Lucas numbers as L(0, n) = Lucas(n), where Lucas(n) = A000032(n) is extended in the usual way for negative n.

The Fibonacci string of length n is defined Fibonacci(n) = cat(Fibonacci(n - 1), Fibonacci(n - 2)) for 1 < n and the initial conditions Fibonacci(0) = "1" and Fibonacci(1) = "0", where 'cat' is the operation of concatenating strings. The length of Fibonacci(n) is A352744(1, n). The sequence starts: "1", "0", "01", "010", "01001", "01001010", ...

Apart from the first four terms seems to be identical with A117248.