A066853 -id:A066853 - cp's OEIS frontend

A248740 a(n) = Fibonacci(n) mod 1000.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 597, 584, 181, 765, 946, 711, 657, 368, 25, 393, 418, 811, 229, 40, 269, 309, 578, 887, 465, 352, 817, 169, 986, 155, 141, 296, 437, 733, 170, 903, 73, 976, 49, 25, 74, 99, 173, 272

Offset: 0

Franz Vrabec, Oct 13 2014

nonn

The sequence is periodic with period 1500 = A001175(1000).
A number m of {0, 1, ..., 999} is not in the range of this sequence, iff m is congruent to 4 or 6 mod 8.
These numbers are the 250 = 1000 - A066853(1000) numbers of the set {4, 6, 12, 14, ..., 996, 998}. E.g., a Fibonacci number will never end in the digits '100'.

			a(17) = (a(16) + a(15)) mod 1000 = (987 + 610) mod 1000 = 1597 mod 1000 = 597.

Alois P. Heinz, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, order 1500.

Cf. A000045, A003893, A105471.

[Fibonacci(n) mod 1000: n in [0..80]]; // Vincenzo Librandi, Oct 17 2014

a:= proc(n) option remember;
      `if`(n<2, n, irem(a(n-1)+a(n-2), 1000))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 18 2015

vector(100,n,fibonacci(n-1)%1000) \\ Derek Orr, Oct 17 2014

a(n) = (a(n-1) + a(n-2)) mod 1000 for n>1, a(0) = 0, a(1) = 1.

More terms from Vincenzo Librandi, Oct 17 2014

A249104 Defective numbers: A complete residue system mod a(n) does not exist in the Fibonacci sequence.

Original entry on oeis.org

8, 11, 12, 13, 16, 17, 18, 19, 21, 22, 23, 24, 26, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87

Offset: 1

Franz Vrabec, Oct 21 2014

nonn

Every multiple of a member is a member.
Every integer 2^r*m (r>2, m odd) is a member.
Every prime p congruent 1, 9, 11, 13, 17, 19 (mod 20) is a member (see reference).

			16 is a member because A066853(16) = 11 < 16.

A. P. Shah, Fibonacci Sequence Modulo m, Fibonacci Quarterly, Vol.6, No.2 (1968), 139-141.

Cf. A000045, A066853.

Complement of A079002. - Jeppe Stig Nielsen, Dec 11 2017

isok(k) = {if(k<8, return(0)); my(v=List([1, 2])); while(v[#v]!=1 || v[#v-1]!=0, listput(v, (v[#v]+v[#v-1])%k)); #Set(v)Jinyuan Wang, Mar 20 2020

A066853(a(n)) < a(n) in ascending order.

More terms from Jinyuan Wang, Mar 20 2020

cp's OEIS Frontend

A248740 a(n) = Fibonacci(n) mod 1000.

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