OFFSET
1,3
COMMENTS
The length of row n >= 2 is (prime(n)-1)/2 = A005097(n-1), and for row n = 1 it is 1.
The other roots of x^2 + b modulo prime(n) are given in A269597.
See A269595 for the irregular triangle with the quadratic residues -b modulo prime(n) = A000040(n), for n >= 1. For n=1 (prime 2) there is a double root x1 = x2 = 1 of x^2 + 1 (mod 2).
Each row n >= 2 consists of a certain permutation of 1, 2, ..., (prime(n)-1)/2.
For a(n), n >= 2, see column x_1 of the table in the Wolfdieter Lang link.
LINKS
FORMULA
EXAMPLE
The irregular triangle T(n, k) begins (P(n) stands here for prime(n)):
n, P(n)\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1, 2: 1
2, 3: 1
3, 5: 2 1
4, 7: 2 3 1
5, 11: 3 4 2 5 1
6: 13: 5 6 3 2 4 1
7, 17: 4 7 8 3 5 2 6 1
8, 19: 6 4 7 3 8 5 9 2 1
9, 23: 8 4 6 9 3 10 11 2 7 5 1
10, 29: 12 5 13 9 14 7 4 10 3 6 8 11 2 1
...
Row n=7 (prime 17) is the permutation (in cycle notation) (1,4,3,8)(2,7,6) of {1, 2, ..., 8}.
MATHEMATICA
nn = 12; s = Table[Select[Range[Prime@ n - 1], JacobiSymbol[#, Prime@ n] == 1 &], {n, nn}]; t = Table[Prime@ n - s[[n, (Prime@ n - 1)/2 - k + 1]], {n, Length@ s}, {k, (Prime@ n - 1)/2}] /. {} -> {1};
Prepend[Table[SelectFirst[Range@ #, Function[x, Mod[x^2 + t[[n, k]], #] == 0]] &@ Prime@ n, {n, 2, Length@ t}, {k, (Prime@ n - 1)/2}], {1}] // Flatten (* Michael De Vlieger, Apr 04 2016, Version 10 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Wolfdieter Lang, Apr 03 2016
STATUS
approved