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%I #11 Jun 03 2024 08:54:21
%S 1,1,1,1,1,1,1,2,3,1,1,2,1,3,1,1,2,2,3,3,1,1,2,0,1,0,3,1,1,1,1,1,1,1,
%T 1,1,1,2,2,3,1,2,3,3,1,1,2,0,3,2,3,2,0,3,1,1,2,2,1,0,1,0,1,3,3,1,1,2,
%U 2,1,0,2,3,0,1,3,3,1,1,2,3,3,2,0,1,0,3,2,2,3,1
%N Triangle read by rows: T(n, k) = [n - k + 1 | k] where [n | k] is defined below.
%H Carl Friedrich Gauss, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN373456743%7CLOG_0008">Vierter Abschnitt. Von den Congruenzen zweiten Grades. Quadratische Reste und Nichtreste. Art. 97</a>, in "Untersuchungen über die höhere Arithmetik", Hrsg. H. Maser, Verlag von Julius Springer, Berlin, 1889.
%F Let two positive numbers n, k be given. We write (n R k) if two integers x and y exist, such that x^2 = n + k*y, and (n N k) otherwise. If the condition is satisfied n is called a quadratic residue modulo k. We distinguish four cases:
%F [n | k] := 0 if (n N k) and (k N n);
%F [n | k] := 1 if (n R k) and (k R n);
%F [n | k] := 2 if (n R k) and (k N n);
%F [n | k] := 3 if (n N k) and (k R n).
%F We set T(n, k) = [n - k + 1 | k].
%F Exchanging 2 <-> 3 reverses the rows.
%F All terms of row n are 1 <==> n = 1, 2 or n is of the form k*(k-2), k >= 3.
%e Triangle starts:
%e [ 1] 1;
%e [ 2] 1, 1;
%e [ 3] 1, 1, 1;
%e [ 4] 1, 2, 3, 1;
%e [ 5] 1, 2, 1, 3, 1;
%e [ 6] 1, 2, 2, 3, 3, 1;
%e [ 7] 1, 2, 0, 1, 0, 3, 1;
%e [ 8] 1, 1, 1, 1, 1, 1, 1, 1;
%e [ 9] 1, 2, 2, 3, 1, 2, 3, 3, 1;
%e [10] 1, 2, 0, 3, 2, 3, 2, 0, 3, 1;
%e [11] 1, 2, 2, 1, 0, 1, 0, 1, 3, 3, 1;
%e [12] 1, 2, 2, 1, 0, 2, 3, 0, 1, 3, 3, 1;
%p QRS := proc(n, k) local QR, p, q, a, b;
%p QR := (a, n) -> NumberTheory:-QuadraticResidue(a, n);
%p a := QR(n, k); b := QR(k, n);
%p if a = -1 and b = -1 then return 0 fi;
%p if a = 1 and b = 1 then return 1 fi;
%p if a = 1 and b = -1 then return 2 fi;
%p if a = -1 and b = 1 then return 3 fi;
%p end: for n from 1 to 12 do lprint([n], seq(QRS(n-k+1, k), k = 1..n)) od;
%o (Python)
%o from sympy.ntheory import is_quad_residue
%o def QR(n, k): return is_quad_residue(n, k)
%o def QRS(n, k):
%o a = QR(n, k); b = QR(k, n)
%o if not a and not b: return 0
%o if a and b: return 1
%o if a and not b: return 2
%o if not a and b: return 3
%o def T(n, k): return QRS(n - k + 1, k)
%o for n in range(1, 13): print([n], [T(n, k) for k in range(1, n + 1)])
%Y Cf. A373223, A373355 (restricted to primes).
%K nonn,tabl
%O 1,8
%A _Peter Luschny_, Jun 02 2024
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