This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181190 #19 Jan 25 2017 04:26:59 %S A181190 1,60,124,1560,4686,1456,18744,585936,4882810,212784 %N A181190 Maximal length of chain-addition sequence mod 10 with window of size n. %C A181190 Chain addition mod 10 with window n: take an n-digit 'seed'. Take the sum of its digits mod 10 and append to the seed. Repeat with the last n digits of the string, until the seed appears again. %C A181190 This sequence shows the lengths of the longest sequences for different window sizes. %C A181190 a(1)-a(10) all occur for seed 1 (among others). If this is always true, the sequence continues: 406224, 12695306, 4272460934, 380859180, 122070312496, 518798826, 3433227539058. - _Lars Blomberg_, Feb 12 2013 %C A181190 Comment from _Michel Lagneau_, Jan 20 2017, edited by _N. J. A. Sloane_, Jan 24 2017: (Start) %C A181190 If seed 1 is always as good as or better than any other, as seems likely, then this sequence has the following alternative description. %C A181190 Consider the n initial terms of an infinite sequence S(k, n) of decimal digits given by 0, 0,..., 0, 1. The succeeding terms are given by the final digits in the sum of the n immediately preceding terms. The sequence lists the period of each sequence corresponding to n = 2, 3, ... %C A181190 a(2) = period of A000045 mod 10 (Fibonacci numbers mod 10) = A001175(10). %C A181190 a(3) = period of A000073 mod 10 (tribonacci numbers mod 10) = A046738(10). %C A181190 a(4) = period of A000078 mod 10 (tetranacci numbers mod 10) = A106295(10). %C A181190 a(5) = period of A001591 mod 10 (pentanacci numbers mod 10) = A106303(10). %C A181190 a(6) = period of A001592 mod 10 (hexanacci numbers mod 10). %C A181190 a(7) = period of A122189 mod 10 (heptanacci numbers mod 10). %C A181190 a(8) = period of A079262 mod 10 (octanacci numbers mod 10). %C A181190 a(4) = 1560 because the four initial terms 0, 0, 0, 1 => S(k, 4) = 0, 0, 0, 1, 1, 2, 4, 8, 5, 9, 6, 8, 8, 1, 3, 0, 2, 6, 1, 9, 8, ... (tetranacci numbers mod 10). This sequence is periodic with period 1560: %C A181190 S(1560 + 1, 4) = S(1, 4) = 0, %C A181190 S(1560 + 2, 4) = S(2, 4) = 0, %C A181190 S(1560 + 3, 4) = S(3, 4) = 0, %C A181190 S(1560 + 4, 4) = S(4, 4) = 1. %C A181190 (End) %e A181190 For n=2, the longest sequence begins with '01' (among others): %e A181190 01123583145943707741561785381909987527965167303369549325729101. %e A181190 It is 60 digits long (not counting the second '01' at the end). %e A181190 For n=3, one of the longest sequences begins again with '001': %e A181190 00112473441944756893025746770415061742394699425184352079627546556679289964992013 %e A181190 48570291225960516297849144970639807524172091001 (124 digits long without the second '001'). %Y A181190 Cf. A000045, A000073, A000078, A001591, A001592, A003893, A079262, A122189 %K A181190 base,more,nonn %O A181190 1,2 %A A181190 Alexander Dashevsky (atanvarnoalda(AT)gmail.com), Oct 10 2010 %E A181190 a(8)-a(10) from _Lars Blomberg_, Feb 12 2013