A344686 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.

0, 1, -1, 4, -1, -4, 9, 1, -5, -9, 16, 5, -4, -11, -16, 25, 11, -1, -11, -19, -25, 36, 19, 4, -9, -20, -29, -36, 49, 29, 11, -5, -19, -31, -41, -49, 64, 41, 20, 1, -16, -31, -44, -55, -64, 81, 55, 31, 9, -11, -29, -45, -59, -71, -81, 100, 71, 44, 19, -4, -25, -44, -61, -76, -89, -100

Offset: 0

Author

Wolfdieter Lang, Jun 17 2021

Keywords

• sign
• tabl
• easy

Comments

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.)

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.

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

Examples

			The array N2(a, b) begins:
a \ b   0  1  2   3   4   5   6   7   8    9   10 ...
-----------------------------------------------------
O:      0 -1 -4  -9 -16 -25 -36 -49 -64  -81 -100 ...
1:      1 -1 -5 -11 -19 -29 -41 -55 -71  -89 -109 ...
2:      4  1 -4 -11 -20 -31 -44 -59 -76  -95 -116 ...
3:      9  5 -1  -9 -19 -31 -45 -61 -79  -99 -121 ..
4:     16 11  4  -5 -16 -29 -44 -61 -80 -101 -124 ...
5:     25 19 11   1 -11 -25 -41 -59 -79 -101 -125 ...
6:     36 29 20   9  -4 -19 -36 -55 -76  -99 -124 ...
7:     49 41 31  19   5 -11 -29 -49 -71  -95 -121 ...
8:     64 55 44  31  16  -1 -20 -41 -64  -89 -116 ...
9:     81 71 59  45  29  11  -9 -31 -55  -81 -109 ...
10:   100 89 76  61  44  25   4 -19 -44  -71 -100 ...
...
------------------------------------------------------
The Triangle T(n, k) begins:
n \ k  0   1   2   3   4   5   6   7   8   9   10 ...
-----------------------------------------------------
O:     0
1:     1  -1
2:     4  -1  -4
3:     9   1  -5  -9
4:    16   5  -4 -11 -16
5:    25  11  -1 -11 -19 -25
6:    36  19   4  -9 -20 -29 -36
7:    49  29  11  -5 -19 -31 -41 -49
8:    64  41  20   1 -16 -31 -44 -55 -64
9:    81  55  31   9 -11 -29 -45 -59 -71 -81
10:  100  71  44  19  -4 -25 -44 -61 -76 -89 -100
...
------------------------------------------------------
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.
Some primes im Q(phi) from  |N(a, -b)| = q, with q a prime in Q:
a = 1:  (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 8), (1, 9), (1, 10), ...
a = 2:  (2, 3), (2, 5), (2, 7), ...
a = 3:  (3, 1), (3, 4), (3, 5), (3, 7), (3, 8), ...
a = 4:  (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), ...
a = 5:  (5, 1), (5, 2), (5, 4), (5, 6), (5, 7), (5, 8), (5, 9), ...
a = 6:  (6, 1), (6, 5), ...
a = 7:  (7, 1), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 8), ...
a = 8:  (8, 3), (8, 7), (8, 9), ...
a = 9:  (9, 1), (9, 2), (9, 4), (9, 5), (9, 7), (9, 10), ...
a = 10: (10, 1), (10, 3), (10, 7), (10, 9), ...
		

References

• F. W. Dodd, Number theory in the quadratic field with golden section unit, Polygonal Publishing House, Passaic, NJ.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Clarendon Press Oxford, 2003.

Crossrefs

Cf. A281385, A344685.

Formula

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

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.

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).