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 A256134 #48 Dec 18 2015 14:46:17 %S A256134 1,1,1,-1,-2,-2,1,3,4,4,5,5,5,-1,-6,-7,-7,-8,-8,-8,1,9,10,10,11,11,12, %T A256134 12,12,-1,-13,-14,-14,-14,1,15,16,16,16,-1,-17,-18,-18,-19,-19,-20, %U A256134 -20,-20,1,21,22,22,23,23,24,24,24,-1,-25,-26,-26,-27,-27,-27,1,28,29,29,29,-1,-30,-31,-31,-31,1,32,33,33,34 %N A256134 The absolute value of a(n) is the length of the n-th line segment of a labyrinth related to odd nonprimes (A014076) and odd primes (A065091) (see Comments lines for definition). %C A256134 In order to construct this sequence we use the following rules: %C A256134 We start with the diagram described in A256253 in which the regions in direction S-W represent the odd nonprimes (A014076) and the regions in direction N-E represent the odd primes (A065091). %C A256134 The diagram must be modified such that the new diagram contains only one region of infinite length as shown in Example section, figure 1. %C A256134 The absolute value of a(n) is the length of the n-th line segment in the walk into the mentioned diagram as shown in Example section, figure 2. %C A256134 The sign of a(n) is the same as the sign of the precedent term in the sequence whose absolute value is 1. %C A256134 The positive value of a(n) means that the line segment rotates in the direction of the clockwise. %C A256134 The negative value of a(n) means that the line segment rotates counter to the clockwise. %C A256134 A line segment of length x can be replaced be x toothpicks with nodes between their endpoints. %C A256134 Also the sequence can be interpreted as an irregular array T(j,k), see Formula section and Example section. %H A256134 Wikipedia, <a href="https://en.wikipedia.org/wiki/Labyrinth">Labyrinth</a> %F A256134 Written as an irregular array we have that: %F A256134 T(1,3) = 1. %F A256134 And for j > 1: %F A256134 T(j,1) = m*(j-1), where m is the precedent term in the sequence whose absolute value is 1. %F A256134 T(j,2) = T(j,1), if 2*j-1 is an odd prime and 2*j+1 is an odd nonprime or if 2*j-1 is an odd nonprime and 2*j+1 is an odd prime. %F A256134 T(j,3) = (-1)*m, if T(j,1) = T(j,2), where m is the precedent term in the sequence whose absolute value is 1, otherwise T(j,3) does not exist. %e A256134 Written as an irregular array T(j,k) the sequence begins: %e A256134 ----------------------- %e A256134 j/k: 1 2 3 %e A256134 ----------------------- %e A256134 1: 1; %e A256134 2: 1, 1, -1; %e A256134 3: -2, -2, 1; %e A256134 4: 3, 4; %e A256134 5: 4, 5; %e A256134 6: 5, 5, -1; %e A256134 7: -6, -7; %e A256134 8: -7, -8; %e A256134 9: -8, -8, 1; %e A256134 10: 9, 10; %e A256134 11: 10, 11; %e A256134 12: 11, 12; %e A256134 13: 12, 12, -1; %e A256134 14: -13, -14; %e A256134 15: -14, -14, 1; %e A256134 16: 15, 16; %e A256134 17: 16, 16; -1; %e A256134 18: -17, -18; %e A256134 19: -18, -19: %e A256134 20: -19, -20; %e A256134 ... %e A256134 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A256134 . | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | 37 %e A256134 . | | | _ _ _ _ _ _ _ _ _ _ _ _ _ _ | | 31 %e A256134 . | | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ | | | 29 %e A256134 . | | | | | _ _ _ _ _ _ _ _ _ _ | | | | 23 %e A256134 . | | | | | | | _ _ _ _ _ _ _ _ | | | | | 19 %e A256134 . | | | | | | |_ _ _ _ _ _ _ _ | | | | | | 17 %e A256134 . | | | | | | | _ _ _ _ _ _ | | | | | | | 13 %e A256134 . | | | | | | | | _ _ _ _ | | | | | | | | 11 %e A256134 . | | | | | | | | | _ _ | | | | | | | | | 7 %e A256134 . | | | | | | | | |_ _ | | | | | | | | | | 5 %e A256134 . A014076 | | | | | | | | | | | | | | | | | | | | 3 %e A256134 . 1 | | | | | | | | |_|_ _| | | | | | | | | | A065091 %e A256134 . 9 | | | | | | | |_ _ _ _ _|_ _| | | | | | | %e A256134 . 15 | | | | | | |_ _ _ _ _ _ _ _ _| | | | | | %e A256134 . 21 | | | | | |_ _ _ _ _ _ _ _ _ _ _| | | | | %e A256134 . 25 | | | | |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | %e A256134 . 27 | | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | %e A256134 . 33 | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A256134 . 35 | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| %e A256134 . 39 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A256134 . %e A256134 Figure 1. Here the diagram described in A256253 was modified such that the new diagram contains only one region of infinite length. %e A256134 . %e A256134 Illustration of initial terms (n = 1..46): %e A256134 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A256134 . | _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ | %e A256134 . | | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ | | %e A256134 . | | | _ _ _ _ _ _ _ _ _ _ _ _ | | | %e A256134 . | | | | | | _ _ _ _ _ _ _ _ _ | | | | %e A256134 . | | | | | | |_ _ _ _ _ _ _ _ | | | | | %e A256134 . | | | | | | _ _ _ _ _ _ _ | | | | | | %e A256134 . | | | | | | | _ _ _ _ _ | | | | | | | %e A256134 . | | | | | | | | _ _ _ | | | | | | | | %e A256134 . | | | | | | | | |_ _ | | | | | | | | | %e A256134 . | | | | | | | | _ | | | | | | | | | | %e A256134 . | | | | | | | | | |_| | | | | | | | | | %e A256134 . | | | | | | | |_ _ _ _| |_| | | | | | | %e A256134 . | | | | | | |_ _ _ _ _ _ _ _| | | | | | %e A256134 . | | | | | |_ _ _ _ _ _ _ _ _ _| | | | | %e A256134 . | | | | |_ _ _ _ _ _ _ _ _ _ _ _| | | | %e A256134 . | | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| | | %e A256134 . | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | %e A256134 . | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| Labyrinth %e A256134 . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ <-- entrance %e A256134 . %e A256134 Figure 2. Interpreted as a sequence, the absolute value of a(n) is the length of the n-th line segment starting from the center of the structure. The figure shows the first 46 line segments. Note that the structure looks like a labyrinth. %Y A256134 Cf. A005408, A014076, A065091, A256253. %K A256134 sign,tabf,walk,look %O A256134 1,5 %A A256134 _Omar E. Pol_, Mar 31 2015