OFFSET
0
COMMENTS
See A329593.
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.
REFERENCES
Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics; Cambridge University Press, 2012 (third printing).
LINKS
Wolfdieter Lang, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, On the Equivalence of Three Complete Cyclic Systems of Integers, arXiv:2008.04300 [math.NT], 2020.
FORMULA
a(n) = +1 if 2^A003558(n) - 1 == 0 (mod (2*n+1)), otherwise -1, for n >= 0.
MAPLE
spow := proc(a, n) numtheory:-mlog(-1, a, n);
if % = FAIL then numtheory:-order(a, n) else % fi; a^% end:
A332433 := n -> `if`(n=0, 1, mods(spow(2, 2*n+1), 2*n+1)):
seq(A332433(n), n=0..70); # Peter Luschny, Mar 11 2020
MATHEMATICA
Suborder[a_, n_] := If[n > 1 && GCD[a, n] == 1,
Min[MultiplicativeOrder[a, n, {-1, 1}]], 0];
A003558[n_] := If[n == 1, 1, Suborder[2, 2n+1]];
a[n_] := If[n == 0, 1, If[PowerMod[2, A003558[n], 2n+1] - 1 == 0, 1, -1]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Oct 17 2024 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Wolfdieter Lang, Feb 13 2020
STATUS
approved