A130893 - OEIS

%I #77 Oct 19 2024 15:57:32

%S 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 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 6,3,9,2,1,3,4,7,1,8,9,7,6,3,9,2,1,3,4

%N Lucas numbers (beginning with 1) mod 10.

%C Period 12: repeat [1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9, 2].

%H G. C. Greubel, <a href="/A130893/b130893.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 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 <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 a(n) = (a(n-2) + a(n-1)) mod 10, with a(0) = 1, a(1) = 3.

%F a(n) = A000204(n+1) mod 10 = A000032(n+1) mod 10. - _Joerg Arndt_, Sep 17 2013

%F 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 From _G. C. Greubel_, Feb 08 2016: (Start)

%F 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 a(n+12) = a(n). (End)

%e 1 + 3 = 4 = 4 mod 10, then a(3) = 4.

%e 3 + 4 = 7 = 7 mod 10, then a(4) = 7.

%e 4 + 7 = 11 = 1 mod 10, then a(5) = 1.

%t Nest[Append[#, Mod[Total[Take[#, -2]], 10]] &, {1, 3}, 110]  (* _Harvey P. Dale_, Apr 05 2011 *)

%t t = {1, 3}; Do[AppendTo[t, Mod[t[[-1]] + t[[-2]], 10]], {99}]; t (* _T. D. Noe_, Sep 16 2013 *)

%t Mod[LucasL[Range[100]], 10] (* _Alonso del Arte_, Sep 30 2015 *)

%t LinearRecurrence[{1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1}, {1, 3, 4, 7,

%t   1, 8, 9, 7, 6, 3, 9}, 100] (* _G. C. Greubel_, Feb 08 2016 *)

%o (Ruby)

%o def truncM10(n)

%o ..a = 1

%o ..b = 3

%o ..n.times do

%o ....a, b = (b % 10), ((a + b) % 10)

%o ..end

%o ..return b

%o end

%o # _Joseph P. Shoulak_, Sep 15 2013

%o (PARI) a(n) = (fibonacci(n+1)+fibonacci(n-1)) % 10;

%o vector(100, n, a(n)) \\ _Altug Alkan_, Sep 30 2015

%o (Magma) [Lucas(n) mod 10: n in [1..100]]; // _Vincenzo Librandi_, Oct 01 2015

%Y Cf. A000032, A003983, A111958.

%K easy,nonn,base

%O 0,2

%A Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 22 2007

%E Corrected and extended by _Harvey P. Dale_, Apr 05 2011

%E New name from _Joerg Arndt_, Sep 17 2013