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Signs appearing in the definition of A329593.
9

%I #29 Oct 17 2024 16:43:42

%S 1,-1,-1,1,-1,-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,1,-1,1,-1,-1,1,1,1,1,-1,

%T 1,-1,-1,-1,1,-1,-1,1,1,1,1,1,1,-1,-1,1,1,1,1,1,1,-1,-1,-1,1,1,-1,-1,

%U 1,-1,1,1,1,-1,1,-1,1,-1,-1,1,1,-1,-1,1

%N Signs appearing in the definition of A329593.

%C See A329593.

%C a(n), for n >= 1, gives the parity (even if -1, odd if 1) of the number of columns of each coach of the complete coach system Sigma(2*n+1); see the quasi-order theorem of Hilton and Pedersen, p. 261 (where r should be r_i, for any i from {1,2,..., c(2*n+1)= A135303(n)}). See also the array A332434 for the coach lengths.

%D Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics; Cambridge University Press, 2012 (third printing).

%H Wolfdieter Lang, <a href="/A332433/b332433.txt">Table of n, a(n) for n = 0..1000</a>

%H Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.

%F a(n) = +1 if 2^A003558(n) - 1 == 0 (mod (2*n+1)), otherwise -1, for n >= 0.

%p spow := proc(a, n) numtheory:-mlog(-1, a, n);

%p if % = FAIL then numtheory:-order(a, n) else % fi; a^% end:

%p A332433 := n -> `if`(n=0, 1, mods(spow(2, 2*n+1), 2*n+1)):

%p seq(A332433(n), n=0..70); # _Peter Luschny_, Mar 11 2020

%t Suborder[a_, n_] := If[n > 1 && GCD[a, n] == 1,

%t Min[MultiplicativeOrder[a, n, {-1, 1}]], 0];

%t A003558[n_] := If[n == 1, 1, Suborder[2, 2n+1]];

%t a[n_] := If[n == 0, 1, If[PowerMod[2, A003558[n], 2n+1] - 1 == 0, 1, -1]];

%t Table[a[n], {n, 0, 70}] (* _Jean-François Alcover_, Oct 17 2024 *)

%Y Cf. A003558, A135303, A329593, A332434.

%K sign

%O 0

%A _Wolfdieter Lang_, Feb 13 2020