%I #10 Feb 13 2018 00:19:47

%S 1,1,2,1,2,3,1,3,4,2,5,1,5,6,3,2,4,1,4,7,8,3,5,2,6,1,6,4,7,3,8,5,9,2,

%T 1,8,4,6,9,3,10,11,2,7,5,1,12,5,13,9,14,7,4,10,3,6,8,11,2,1

%N Irregular triangle giving in row n the smaller of the two roots x1 of x^2 + b modulo prime(n) from {0, 1, ..., prime(n)-1} corresponding to b from row n of A269595.

%C The length of row n >= 2 is (prime(n)-1)/2 = A005097(n-1), and for row n = 1 it is 1.

%C The other roots of x^2 + b modulo prime(n) are given in A269597.

%C 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).

%C Each row n >= 2 consists of a certain permutation of 1, 2, ..., (prime(n)-1)/2.

%C For a(n), n >= 2, see column x_1 of the table in the Wolfdieter Lang link.

%H Wolfdieter Lang, <a href="/A268922/a268922.pdf">Note on a Recurrence for Approximation Sequences of p-adic Square Roots</a>

%F T(n, k) gives the smaller zero of x^2 + A269595(n, k) == 0 (mod prime(n)), n >= 1, for k=1 if n=1 and k = 1, 2, ..., (prime(n)-1)/2 = A005097(n-1) for n >= 2. Representatives are taken from the complete residue class {0, 1 ,..., prime(n)-1}.

%F T(n, k) = prime(n) - A269597(n, k).

%e The irregular triangle T(n, k) begins (P(n) stands here for prime(n)):

%e n, P(n)\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14

%e 1, 2: 1

%e 2, 3: 1

%e 3, 5: 2 1

%e 4, 7: 2 3 1

%e 5, 11: 3 4 2 5 1

%e 6: 13: 5 6 3 2 4 1

%e 7, 17: 4 7 8 3 5 2 6 1

%e 8, 19: 6 4 7 3 8 5 9 2 1

%e 9, 23: 8 4 6 9 3 10 11 2 7 5 1

%e 10, 29: 12 5 13 9 14 7 4 10 3 6 8 11 2 1

%e ...

%e Row n=7 (prime 17) is the permutation (in cycle notation) (1,4,3,8)(2,7,6) of {1, 2, ..., 8}.

%t 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};

%t 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 *)

%Y Cf. A000040, A005097, A269595, A269597 (x2).

%K nonn,tabf,easy

%O 1,3

%A _Wolfdieter Lang_, Apr 03 2016