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%I #17 Feb 10 2022 13:44:24
%S 1,0,2,1,4,0,1,2,4,1,8,2,1,0,8,1,2,0,1,4,2,1,8,4,1,2,8,1,16,2,1,4,16,
%T 1,2,4,1,0,2,1,16,0,1,2,16,1,4,2,1,0,4,1,2,0,1,8,2,1,4,8,1,2,4,1,16,2,
%U 1,8,16,1,2,8,1,4,2,1,16,4,1,2,16,1,32,2,1
%N a(1) = 0; for m >= 0, a(3m) = 1; for m >= 1, a(3m-1) = 2*a(m-1), a(3m+1) = 2*a(m+1).
%C The sequence extends to negative n by defining a(n) = a(-n).
%C Consider a Sierpinski arrowhead curve formed of unit edges numbered consecutively from 0 at its axis of symmetry and aligned with an infinite Sierpinski gasket so that each edge is contained in the boundary of either the plane sector occupied by the gasket or a triangular region of the gasket's complement. If a(n) = 0, the n-th edge is contained in the sector boundary, otherwise the relevant triangular region has side a(n). Every length 3 segment of these boundaries contains exactly one edge of the arrowhead curve. A191108 lists positive n such that edge n is contained in the plane sector boundary. See A307672 for further properties.
%C See the graphic (in the links) of the arrowhead curve aligned with the gasket. Note the even-indexed edges (colored red) are the edges contained in a triangular region boundary on the left side of the vector. - _Peter Munn_, Jul 29 2019
%C a(n) = 0 if A307744(n) = 0, otherwise a(n) = 2^(A307744(n) - 1). Thus, this sequence has the symmetries of A307744, which are similar to ruler functions (especially A051064) and described further in A307744.
%C When listening to this, set pitch modulus to 35 or 36.
%H Peter Munn, <a href="/A309054/b309054.txt">Table of n, a(n) for n = 0..4374</a>
%H Peter Munn, <a href="/A309054/a309054_1.png">Arrowhead curve aligned with Sierpiński gasket</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpi%C5%84ski_arrowhead_curve">Sierpiński arrowhead curve</a>
%F a(n) = 0 if A307744(n) = 0, otherwise a(n) = 2^(A307744(n) - 1).
%F a(A191108(i)) = 0 for i >= 1.
%F if a(n) = 2^k, a(3^(k+1)+n) = a(3^(k+1)-n) = 2^k.
%F a((m-1)*3^k + 1) = a((m+1)*3^k - 1) for k >= 1, all integer m.
%F Upper bound relations: (Start)
%F for k >= 1, let m_k = A034472(k-1) = 3^(k-1)+1.
%F a(n) < 2^k, for -m_k < n < m_k.
%F a(-m_k) = a(m_k) = 2^k.
%F (End)
%F Sum_{n=-3^k..3^k-1} (a(n) + 1) = 3 * 4^k.
%F Sum_{n=-3m..3m-1} (a(n) + 1) = 4 * Sum_{n=-m..m-1} (a(n) + 1) (conjectured).
%e As 4 is congruent to 1 modulo 3, a(4) = a(3*1 + 1) = 2*a(1+1) = 2*a(2).
%e As 2 is congruent to -1 modulo 3, a(2) = a(3*1 - 1) = 2*a(1-1) = 2*a(0).
%e As 0 is congruent to 0 modulo 3, a(0) = 1. So a(2) = 2*a(0) = 2*1 = 2. So a(4) = 2*a(2) = 2*2 = 4.
%Y Cf. A034472, A051064, A191108, A307672, A307744.
%K nonn,look,hear
%O 0,3
%A _Peter Munn_, Jul 09 2019
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