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

Author

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

Keywords

• nonn
• base
• easy

Comments

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.

References

• 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.

Links

• 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).

Crossrefs

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

Cf. A008963, A261606.

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

More terms from Ray Chandler, Nov 15 2003