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

%I A003893 #91 Apr 15 2025 15:49:07
%S A003893 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,
%T A003893 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,
%U A003893 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
%N A003893 a(n) = Fibonacci(n) mod 10.
%C A003893 All blocks of 60 successive terms contain 20 even and 40 odd numbers. - _Reinhard Zumkeller_, Apr 09 2005
%C A003893 These are the analogs of the Fibonacci numbers in carryless arithmetic mod 10.
%D A003893 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.
%H A003893 Vincenzo Librandi, <a href="/A003893/b003893.txt">Table of n, a(n) for n = 0..10000</a>
%H A003893 David Applegate, Marc LeBrun and N. J. A. Sloane, <a href="http://neilsloane.com/doc/carry1.pdf">Carryless Arithmetic (I): The Mod 10 Version</a>.
%H A003893 H. S. M. Coxeter, <a href="/A000201/a000201.pdf">The Golden Section, Phyllotaxis and Wythoff's Game</a>, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]
%H A003893 Gregory P. Dresden, <a href="https://dresden.academic.wlu.edu/files/2017/08/lnzd2.pdf">Three transcendental numbers from the last non-zero digits of n^n, F_n and n!</a>, Math. Mag., pp. 96-105, vol. 81, 2008.
%H A003893 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html">The Mathematical Magic of the Fibonacci Numbers</a>.
%H A003893 Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, <a href="https://doi.org/10.1609/aaai.v39i2.32162">Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences</a>, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
%H A003893 <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%H A003893 <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%H A003893 <a href="/index/Rec#order_39">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A003893 Periodic with period 60 = A001175(10).
%F A003893 From _Reinhard Zumkeller_, Apr 09 2005: (Start)
%F A003893 a(n) = (a(n-1) + a(n-2)) mod 10 for n > 1, a(0) = 0, a(1) = 1.
%F A003893 a(n) = A105471(n) - A105472(n)*10 = A105471(n)/10. (End)
%F A003893 a(n) = A010879(A000045(n)). - _Michel Marcus_, Nov 19 2022
%p A003893 with(combinat,fibonacci); A003893 := proc(n) fibonacci(n) mod 10; end;
%t A003893 Table[Mod[Fibonacci[n], 10], {n, 0, 99}] (* _Alonso del Arte_, Jul 29 2013 *)
%t A003893 Table[IntegerDigits[Fibonacci[n]][[-1]], {n, 0, 99}] (* _Alonso del Arte_, Jul 29 2013 *)
%t A003893 NumberDigit[Fibonacci[Range[0,120]],0] (* Requires Mathematica version 12 or later *) (* _Harvey P. Dale_, Jul 05 2021 *)
%o A003893 (Haskell)
%o A003893 a003893 n = a003893_list !! n
%o A003893 a003893_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 10)
%o A003893                        (tail a003893_list) a003893_list
%o A003893 -- _Reinhard Zumkeller_, Jul 01 2013
%o A003893 (PARI) a(n)=fibonacci(n)%10 \\ _Charles R Greathouse IV_, Feb 03 2014
%o A003893 (Magma) [Fibonacci(n) mod 10: n in [0..100]]; // _Vincenzo Librandi_, Feb 04 2014
%o A003893 (Python)
%o A003893 A003893_list, a, b, = [], 0, 1
%o A003893 for _ in range(10**3):
%o A003893     A003893_list.append(a)
%o A003893     a, b = b, (a+b) % 10 # _Chai Wah Wu_, Nov 26 2015
%Y A003893 Cf. A000045, A010879, A089911, A105471, A105472.
%Y A003893 Cf. A008963, A261606.
%K A003893 nonn,base,easy
%O A003893 0,4
%A A003893 _N. J. A. Sloane_, elipper(AT)uoft02.utoledo.edu
%E A003893 More terms from _Ray Chandler_, Nov 15 2003