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 A337713 Irregular triangle T read by rows: row n gives the inverse elements of row n of A216319 Modd(n), for n >= 1. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=1..103. 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 Cf. A055034, A203571, A187360, A216319. Sequence in context: A250207 A216319 A355001 * A309425 A218355 A103790 Adjacent sequences: A337710 A337711 A337712 * A337714 A337715 A337716 KEYWORD nonn,tabf,easy AUTHOR Wolfdieter Lang, Oct 20 2020 STATUS approved

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Last modified February 22 06:21 EST 2024. Contains 370240 sequences. (Running on oeis4.)