A002752 -id:A002752 - cp's OEIS frontend

A002708 a(n) = Fibonacci(n) mod n.

0, 1, 2, 3, 0, 2, 6, 5, 7, 5, 1, 0, 12, 13, 10, 11, 16, 10, 1, 5, 5, 1, 22, 0, 0, 25, 20, 11, 1, 20, 1, 5, 13, 33, 30, 0, 36, 1, 37, 35, 1, 34, 42, 25, 20, 45, 46, 0, 36, 25, 32, 23, 52, 8, 5, 21, 40, 1, 1, 0, 1, 1, 43, 59, 60, 52, 66, 65, 44, 15, 1, 0, 72, 73, 50, 3, 2, 44, 1, 5, 7, 1, 82, 24

Offset: 1

John C. Hallyburton, Jr. (hallyb(AT)evms.ENET.dec.com)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 891.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)

Cf. A002726, A002752, A023172 (indices of 0's), A023173 (indices of 1's), A023174-A023182.
Cf. A263101.
Main diagonal of A161553.

[Fibonacci(n) mod n : n in [1..120]]; // Vincenzo Librandi, Nov 19 2015

with(combinat): [ seq( fibonacci(n) mod n, n=1..80) ];
# second Maple program:
a:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:= r.M mod n fi;
         if p=0 then break fi; M:= M.M mod n
      od; r[1, 2]
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 26 2016

Table[Mod[Fibonacci[n], n], {n, 1, 100}] (* Stefan Steinerberger, Apr 18 2006 *)

a(n) = fibonacci(n) % n; \\ Michel Marcus, May 11 2016

A002708_list, a, b, = [], 1, 1
for n in range(1,10**4+1):
    A002708_list.append(a%n)
    a, b = b, a+b # Chai Wah Wu, Nov 26 2015

More terms from Stefan Steinerberger, Apr 18 2006

A352747 Array read by ascending antidiagonals. A(n, k) = F(k, n) mod n for n >= 1 and k >= 0, where F(n, k) = A352744(n, k) are the Fibonacci numbers, A(0, k) = 1 for k >= 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 1, 3, 1, 2, 0, 0, 1, 5, 3, 0, 1, 1, 0, 1, 1, 1, 3, 3, 0, 0, 0, 1, 5, 0, 3, 3, 2, 2, 1, 0, 1, 3, 2, 6, 5, 3, 1, 1, 0, 0, 1, 4, 1, 7, 5, 1, 3, 0, 0, 1, 0, 1, 0, 9, 8, 4, 4, 3, 3, 3, 2, 0, 0, 1, 5, 1, 4, 6, 1, 3, 5, 3, 2, 1, 1, 0, 1

Offset: 0

Peter Luschny, Apr 08 2022

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).

			Array starts (periods are indicated with () ):
[n\k] 0   1   2   3   4  5  6   7   8   9  10  11  12
----------------------------------------------------------
[ 0] (1), 1,  1,  1,  1, 1, 1,  1,  1,  1,  1,  1,  1, ...
[ 1] (0), 0,  0,  0,  0, 0, 0,  0,  0,  0,  0,  0,  0, ...
[ 2] (1,  0), 1,  0,  1, 0, 1,  0,  1,  0,  1,  0,  1, ...
[ 3] (1,  0,  2), 1,  0, 2, 1,  0,  2,  1,  0,  2,  1, ...
[ 4] (2,  1,  0,  3), 2, 1, 0,  3,  2,  1,  0,  3,  2, ...
[ 5] (3), 3,  3,  3,  3, 3, 3,  3,  3,  3,  3,  3,  3, ...
[ 6] (5,  1,  3), 5,  1, 3, 5,  1,  3,  5,  1,  3,  5, ...
[ 7] (1,  0,  6,  5,  4, 3, 2), 1,  0,  6,  5,  4,  3, ...
[ 8] (5,  2,  7,  4,  1, 6, 3,  0), 5,  2,  7,  4,  1, ...
[ 9] (3,  1,  8,  6,  4, 2, 0,  7,  5), 3,  1,  8,  6, ...
[10] (4,  9), 4,  9,  4, 9, 4,  9,  4,  9,  4,  9,  4, ...
[11] (0,  1,  2,  3,  4, 5, 6,  7,  8,  9, 10), 0,  1, ...
[12] (5), 5,  5,  5,  5, 5, 5,  5,  5,  5,  5,  5,  5, ...

Cf. A352744, A002752, A045468, A074215, A088209, A174090, A212804, A270313, A301729, A353280.

f := n -> combinat:-fibonacci(n + 1):
F := proc(n, k) option remember; (n-1)*f(k-1) + f(k) end:
A := (n, k) -> ifelse(n = 0, 1, modp(F(k, n), n)):
for n from 0 to 12 do seq(A(n, k), k = 0..10) od;

F[n_, k_] := (n - 1)*Fibonacci[k] + Fibonacci[k + 1];
A[n_, k_] := If[n == 0, 1, Mod[F[k, n], n]];
Table[A[n, k], {n, 0, 12}, {k, 0, 10}] // TableForm

def F(n, k): return (n - 1)*fibonacci(k) + fibonacci(k + 1)
def A(n,k): return mod(F(k, n), n)
for n in range(13): print([A(n,k) for k in range(13)])

A(n, 0) = A(n, n) = A002752(n).

Clearly 0 <= A(n, k) < n for all k and n > 0.

cp's OEIS Frontend

A002708 a(n) = Fibonacci(n) mod n.

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A352747 Array read by ascending antidiagonals. A(n, k) = F(k, n) mod n for n >= 1 and k >= 0, where F(n, k) = A352744(n, k) are the Fibonacci numbers, A(0, k) = 1 for k >= 0.

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