%I #17 Aug 08 2021 01:41:54

%S 0,1,-1,4,-1,-4,9,1,-5,-9,16,5,-4,-11,-16,25,11,-1,-11,-19,-25,36,19,

%T 4,-9,-20,-29,-36,49,29,11,-5,-19,-31,-41,-49,64,41,20,1,-16,-31,-44,

%U -55,-64,81,55,31,9,-11,-29,-45,-59,-71,-81,100,71,44,19,-4,-25,-44,-61,-76,-89,-100

%N Triangle T(n, k) obtained from the array N2(a, b) = a^2 - a*b - b^2, for a >= 0 and b >= 0, read by upwards antidiagonals.

%C The general array N(a, b) gives the norms of the integers alpha = a*1 + b*phi, for rational integers a and b, with phi = (1 + sqrt(5))/2 = A001622, in the real quadratic number field Q(phi), also called Q(sqrt(5)). N(a, b) := alpha*alpha' = a^2 + a*b - b^2, with alpha' = (a+b)*1 - b*phi. (phi' = (1 - sqrt(5))/2 = 1 - phi.)

%C The present array is N2(a, b) = N(a,-b) = N(-a, b), for a >= 0 and b >= 0. The companion array N1(a, b) = N(a, b), for a >= 0 and b >= 0, is given (as triangle) in A281385.

%C For units and primes in Q(phi), and for references, see A344685.

%D F. W. Dodd, Number theory in the quadratic field with golden section unit, Polygonal Publishing House, Passaic, NJ.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Clarendon Press Oxford, 2003.

%F Array N2(a, b) = a^2 - a*b - b^2, for a >= 0 and b >= 0.

%F Triangle T(n, k) = N2(n-k, k) = N(n-k, -k) = n^2 - 3*n*k + k^2, for n >= 0 and k = 0, 1, ..., n.

%F G.f. for row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k, i.e. of the triangle: G(x, y) = x*(1 - y + (1 -y - 1*y^2)*x - y*(2 - 4*y)*x^2 - y^2*x^3)/((1 - x*y)^3*(1 - x)^3) (compare with the g.f.s in A281385 and A344685).

%e The array N2(a, b) begins:

%e a \ b 0 1 2 3 4 5 6 7 8 9 10 ...

%e -----------------------------------------------------

%e O: 0 -1 -4 -9 -16 -25 -36 -49 -64 -81 -100 ...

%e 1: 1 -1 -5 -11 -19 -29 -41 -55 -71 -89 -109 ...

%e 2: 4 1 -4 -11 -20 -31 -44 -59 -76 -95 -116 ...

%e 3: 9 5 -1 -9 -19 -31 -45 -61 -79 -99 -121 ..

%e 4: 16 11 4 -5 -16 -29 -44 -61 -80 -101 -124 ...

%e 5: 25 19 11 1 -11 -25 -41 -59 -79 -101 -125 ...

%e 6: 36 29 20 9 -4 -19 -36 -55 -76 -99 -124 ...

%e 7: 49 41 31 19 5 -11 -29 -49 -71 -95 -121 ...

%e 8: 64 55 44 31 16 -1 -20 -41 -64 -89 -116 ...

%e 9: 81 71 59 45 29 11 -9 -31 -55 -81 -109 ...

%e 10: 100 89 76 61 44 25 4 -19 -44 -71 -100 ...

%e ...

%e ------------------------------------------------------

%e The Triangle T(n, k) begins:

%e n \ k 0 1 2 3 4 5 6 7 8 9 10 ...

%e -----------------------------------------------------

%e O: 0

%e 1: 1 -1

%e 2: 4 -1 -4

%e 3: 9 1 -5 -9

%e 4: 16 5 -4 -11 -16

%e 5: 25 11 -1 -11 -19 -25

%e 6: 36 19 4 -9 -20 -29 -36

%e 7: 49 29 11 -5 -19 -31 -41 -49

%e 8: 64 41 20 1 -16 -31 -44 -55 -64

%e 9: 81 55 31 9 -11 -29 -45 -59 -71 -81

%e 10: 100 71 44 19 -4 -25 -44 -61 -76 -89 -100

%e ...

%e ------------------------------------------------------

%e Units from norm N(a, -b) = N2(a, b) = +1 or -1, for a >= 0 and b >= 0: +(a, b) or -(a, b), with (a, b) = (0, 1), (1, 0), (1, 1), (2, 1), (3, 2), (5, 3), (8, 5), ...; cases + or - phi^n, n >= 0. Fibonacci neighbors.

%e Some primes im Q(phi) from |N(a, -b)| = q, with q a prime in Q:

%e a = 1: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 8), (1, 9), (1, 10), ...

%e a = 2: (2, 3), (2, 5), (2, 7), ...

%e a = 3: (3, 1), (3, 4), (3, 5), (3, 7), (3, 8), ...

%e a = 4: (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), ...

%e a = 5: (5, 1), (5, 2), (5, 4), (5, 6), (5, 7), (5, 8), (5, 9), ...

%e a = 6: (6, 1), (6, 5), ...

%e a = 7: (7, 1), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 8), ...

%e a = 8: (8, 3), (8, 7), (8, 9), ...

%e a = 9: (9, 1), (9, 2), (9, 4), (9, 5), (9, 7), (9, 10), ...

%e a = 10: (10, 1), (10, 3), (10, 7), (10, 9), ...

%Y Cf. A281385, A344685.

%K sign,tabl,easy

%O 0,4

%A _Wolfdieter Lang_, Jun 17 2021