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A130893 Lucas numbers (beginning with 1) mod 10.

%I A130893 #80 Jul 22 2025 18:20:02
%S A130893 1,3,4,7,1,8,9,7,6,3,9,2,1,3,4,7,1,8,9,7,6,3,9,2,1,3,4,7,1,8,9,7,6,3,
%T A130893 9,2,1,3,4,7,1,8,9,7,6,3,9,2,1,3,4,7,1,8,9,7,6,3,9,2,1,3,4,7,1,8,9,7,
%U A130893 6,3,9,2,1,3,4,7,1,8,9,7,6,3,9,2,1,3,4
%N A130893 Lucas numbers (beginning with 1) mod 10.
%C A130893 Period 12: repeat [1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2].
%H A130893 G. C. Greubel, <a href="/A130893/b130893.txt">Table of n, a(n) for n = 0..10000</a>
%H A130893 Maarten Bullynck, <a href="http://images-archive.math.cnrs.fr/L-histoire-de-l-informatique-et-l-histoire-des-mathematiques-rencontres.html">L’histoire de l’informatique et l’histoire des mathématiques : rencontres, opportunités et écueils</a>, Images des Mathématiques, CNRS, 2015 (in French).
%H A130893 Johann Heinrich Lambert, <a href="http://www.deutschestextarchiv.de/book/show/lambert_architectonic01_1771">Anlage zur Architectonic, oder Theorie des Einfachen und des Ersten in der philosophischen und mathematischen Erkenntniß</a>, 1771.
%H A130893 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
%F A130893 a(n) = (a(n-2) + a(n-1)) mod 10, with a(0) = 1, a(1) = 3.
%F A130893 a(n) = A000204(n+1) mod 10 = A000032(n+1) mod 10. - _Joerg Arndt_, Sep 17 2013
%F A130893 a(n) = f(5(n-1)+2) mod 10, where f(n) is the n-th Fibonacci number (A000045). - _Joseph P. Shoulak_, Sep 15 2013
%F A130893 From _G. C. Greubel_, Feb 08 2016: (Start)
%F A130893 a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - a(n-8) + a(n-9) - a(n-10) + a(n-11).
%F A130893 a(n+12) = a(n). (End)
%e A130893 1 + 3 = 4 = 4 mod 10, then a(3) = 4.
%e A130893 3 + 4 = 7 = 7 mod 10, then a(4) = 7.
%e A130893 4 + 7 = 11 = 1 mod 10, then a(5) = 1.
%t A130893 Nest[Append[#, Mod[Total[Take[#, -2]], 10]] &, {1, 3}, 110]  (* _Harvey P. Dale_, Apr 05 2011 *)
%t A130893 t = {1, 3}; Do[AppendTo[t, Mod[t[[-1]] + t[[-2]], 10]], {99}]; t (* _T. D. Noe_, Sep 16 2013 *)
%t A130893 Mod[LucasL[Range[100]], 10] (* _Alonso del Arte_, Sep 30 2015 *)
%t A130893 LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}, {1, 3, 4, 7,
%t A130893   1, 8, 9, 7, 6, 3, 9}, 100] (* _G. C. Greubel_, Feb 08 2016 *)
%o A130893 (Ruby)
%o A130893 def truncM10(n)
%o A130893   a = 1
%o A130893   b = 3
%o A130893   n.times do
%o A130893     a, b = (b % 10), ((a + b) % 10)
%o A130893   end
%o A130893   return b
%o A130893 end
%o A130893 # _Joseph P. Shoulak_, Sep 15 2013
%o A130893 (PARI) a(n) = (fibonacci(n+1)+fibonacci(n-1)) % 10;
%o A130893 vector(100, n, a(n)) \\ _Altug Alkan_, Sep 30 2015
%o A130893 (Magma) [Lucas(n) mod 10: n in [1..100]]; // _Vincenzo Librandi_, Oct 01 2015
%Y A130893 Cf. A000032, A003983, A111958.
%K A130893 easy,nonn,base
%O A130893 0,2
%A A130893 Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 22 2007
%E A130893 Corrected and extended by _Harvey P. Dale_, Apr 05 2011
%E A130893 New name from _Joerg Arndt_, Sep 17 2013