A105471 -id:A105471 - cp's OEIS frontend

A003893 a(n) = Fibonacci(n) mod 10.

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

Offset: 0

N. J. A. Sloane, elipper(AT)uoft02.utoledo.edu

All blocks of 60 successive terms contain 20 even and 40 odd numbers. - Reinhard Zumkeller, Apr 09 2005

These are the analogs of the Fibonacci numbers in carryless arithmetic mod 10.

G. Marsaglia, The mathematics of random number generators, pp. 73-90 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Math. Mag., pp. 96-105, vol. 81, 2008.
Ron Knott, The Mathematical Magic of the Fibonacci Numbers.
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for sequences related to final digits of numbers
Index entries for sequences related to carryless arithmetic
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0,0,1,0,0,-1,0,0,-1,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1).

Cf. A000045, A010879, A089911, A105471, A105472.

Cf. A008963, A261606.

a003893 n = a003893_list !! n
a003893_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 10)
                       (tail a003893_list) a003893_list
-- Reinhard Zumkeller, Jul 01 2013

[Fibonacci(n) mod 10: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

with(combinat,fibonacci); A003893 := proc(n) fibonacci(n) mod 10; end;

Table[Mod[Fibonacci[n], 10], {n, 0, 99}] (* Alonso del Arte, Jul 29 2013 *)
Table[IntegerDigits[Fibonacci[n]][[-1]], {n, 0, 99}] (* Alonso del Arte, Jul 29 2013 *)
NumberDigit[Fibonacci[Range[0,120]],0] (* Requires Mathematica version 12 or later *) (* Harvey P. Dale, Jul 05 2021 *)

a(n)=fibonacci(n)%10 \\ Charles R Greathouse IV, Feb 03 2014

A003893_list, a, b, = [], 0, 1
for _ in range(10**3):
    A003893_list.append(a)
    a, b = b, (a+b) % 10 # Chai Wah Wu, Nov 26 2015

Periodic with period 60 = A001175(10).
From Reinhard Zumkeller, Apr 09 2005: (Start)
a(n) = (a(n-1) + a(n-2)) mod 10 for n > 1, a(0) = 0, a(1) = 1.
a(n) = A105471(n) - A105472(n)*10 = A105471(n)/10. (End)
a(n) = A010879(A000045(n)). - Michel Marcus, Nov 19 2022

More terms from Ray Chandler, Nov 15 2003

A105472 Next-to-last digit of n-th Fibonacci number in decimal representation, a(n) = 0 for n <= 6.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Apr 09 2005

a(n) = floor(A105471(n)/10) = floor(A000045(n)/10) mod 10;
A105471(n) = a(n)*10 + A003893(n);
the sequence is periodic with period 300; all blocks of 300 successive terms contain 160 even and 140 odd numbers.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 601 terms from Alex Ermolaev)

Cf. A000045, A003893, A105471.

Array[Mod[Floor[Fibonacci[#]/10], 10] &, 105, 0] (* Michael De Vlieger, Jan 04 2018 *)
Join[{0,0,0,0,0,0,0},Table[IntegerDigits[Fibonacci[n]][[-2]],{n,7,120}]] (* Harvey P. Dale, Aug 22 2020 *)

a(n) = (fibonacci(n) % 100)\10; \\ Michel Marcus, Jan 05 2018

A137290 Fibonacci(n) mod 30.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 4, 25, 29, 24, 23, 17, 10, 27, 7, 4, 11, 15, 26, 11, 7, 18, 25, 13, 8, 21, 29, 20, 19, 9, 28, 7, 5, 12, 17, 29, 16, 15, 1, 16, 17, 3, 20, 23, 13, 6, 19, 25, 14, 9, 23, 2, 25, 27, 22, 19, 11, 0, 11, 11, 22, 3, 25, 28, 23, 21, 14, 5, 19, 24, 13, 7, 20, 27, 17, 14

Offset: 1

Aaron M. Churchill (churchil(AT)math.udel.edu), Mar 15 2008

nonn

Has period 120.

Michel Marcus, Table of n, a(n) for n = 1..130
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000045, A003893, A007887, A079345, A105471.

Mod[Fibonacci[Range[80]],30] (* Harvey P. Dale, Sep 12 2022 *)

a(n) = fibonacci(n) % 30 \\ Michel Marcus, Jun 12 2013

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

Original entry on oeis.org

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

cp's OEIS Frontend

A003893 a(n) = Fibonacci(n) mod 10.

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A105472 Next-to-last digit of n-th Fibonacci number in decimal representation, a(n) = 0 for n <= 6.

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A137290 Fibonacci(n) mod 30.

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

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