A337713 Irregular triangle T read by rows: row n gives the inverse elements of row n of A216319 Modd(n), for n >= 1.

1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 5, 3, 1, 5, 3, 7, 1, 7, 5, 1, 7, 3, 9, 1, 7, 9, 3, 5, 1, 5, 7, 11, 1, 9, 5, 11, 3, 7, 1, 9, 11, 3, 5, 13, 1, 13, 11, 7, 1, 11, 13, 9, 7, 3, 5, 15, 1, 11, 7, 5, 15, 3, 13, 9, 1, 7, 5, 13, 11, 17, 1, 13, 15, 11, 17, 7, 3, 5, 9, 1, 13, 17, 9, 11, 3, 7, 19, 1, 17, 19, 13, 5, 11, 1, 15, 9, 19, 5, 17, 3, 13, 7, 21

Offset: 1

Author

Wolfdieter Lang, Oct 20 2020

Keywords

• nonn
• tabf
• easy

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).

Examples

			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.
		

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.

Crossrefs

Cf. A055034, A203571, A187360, A216319.

Programs

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

Formula

T(n, k) = Inverse of A216319(n, k) (Modd n), for n >= 1. For Modd n see the comment above.