OFFSET
1,5
COMMENTS
The length of row n is A055034(n), called here delta(n), for n >= 1.
For the modified modular equivalence relation Modd n see a comment in A203571, and the W. Lang link, Definition 4. p. 25. For Modd(a, n) one has to consider the parity of floor(a/n). If it is even then Modd(a, n) = mod(a, n), otherwise it is mod(-a, n).
The rows of A216319 are the smallest positive restricted residue system mod n with only odd members (RRSodd(n)). This is not a group mod n, but a group Modd n, called here G(rho(n)). This group is isomorphic to the Galois group Gal(Q(rho(n))/Q), where the algebraic number of degree delta(n) is rho(n) = 2*cos(Pi/n), for n >= 1. See A187360 for the minimal polynomials of rho(n), called C(n, x).
LINKS
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012.
FORMULA
T(n, k) = Inverse of A216319(n, k) (Modd n), for n >= 1. For Modd n see the comment above.
EXAMPLE
The irregular triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 ...
1: 1
2: 1
3: 1
4: 1 3
5: 1 3
6: 1 5
7: 1 5 3
8: 1 5 3 7
9: 1 7 5
10: 1 7 3 9
11: 1 7 9 3 5
12: 1 5 7 11
13: 1 9 5 11 3 7
14: 1 9 11 3 5 13
15: 1 13 11 7
16: 1 11 13 9 7 3 5 15
17: 1 11 7 5 15 3 13 9
18: 1 7 5 13 11 17
19: 1 13 15 11 17 7 3 5 9
20: 1 13 17 9 11 3 7 19
...
T(7, 2) = 5 because A216319(7, 2) = 3 and Modd(3*5, 7) = 1 since floor(15/7) = 2 is even, hence Modd(3*5, 7) = mod(15, 7) = 1. The residue classes Modd 7 for 1, 3, 5 are shown in the array given in A113807 (including the negative numbers) [3]*[5] = [1] (Modd 7).
T(9, 2) = 7 because A216319(9, 2) = 5 and Modd(7*5, 9) = 1, since floor(35/9) = 3 is odd, hence Moddn(35, 9) = mod(-35, 9) = 1.
PROG
(PARI) rowa(n) = select(x->(((x%2)==1) && (gcd(n, x)==1)), [1..n]); \\ A216319
Modd(x, n) = if ((x\n)%2, Mod(-x, n), Mod(x, n));
findinvm(k, n) = for (i=1, n, if (Modd(k*i, n) == 1, return(i)));
row(n) = my(ra=rowa(n)); vector(#ra, k, findinvm(ra[k], n)); \\ Michel Marcus, Sep 13 2023
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Wolfdieter Lang, Oct 20 2020
STATUS
approved