A344685 Triangle T(n, k) obtained from the array N1(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, 5, -1, -9, 16, 11, 4, -5, -16, 25, 19, 11, 1, -11, -25, 36, 29, 20, 9, -4, -19, -36, 49, 41, 31, 19, 5, -11, -29, -49, 64, 55, 44, 31, 16, -1, -20, -41, -64, 81, 71, 59, 45, 29, 11, -9, -31, -55, -81, 100, 89, 76, 61, 44, 25, 4, -19, -44, -71, -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*1 + b*phi' = (a+b)*1 - b*phi. (phi' = (1 - sqrt(5))/2 = 1 - phi = -1/phi.)

The present array is N1(a, b) = N(a, b) = N(-a, -b), for a >= 0 and b >= 0. The companion array N2(a, b) = N(a, -b) = N(-a, b), for a >= 0 and b >= 0 is given (as triangle) in A281386.

The subtriangle N(a, b), with 0 <= b <= a, is given in A281385.

The units u = a + b*phi of the integer domain of Q(phi) satisfy N(a, b) = +1 or -1, and they are related to positive and negative integer powers of phi, involving neighboring Fibonacci numbers a and b of different signs. See, e.g., Hardy and Wright, Theorem 257, p. 222 (units are there called unities).

If |N(alpha)| = q, with q a rational prime, then alpha is a prime in Q(phi). See, e.g., the Dodd reference, Theorem 3.4, p. 23. But there are other primes. For all primes see e.g., Hardy and Wright, Theorem 257, p. 222, or Dodd, Theorem 3.10, p. 25. For rational primes which are also primes in Q(phi) (so-called inert primes) see A003631. See the tables in Appendix B of Dodd, pp. 128 - 150, for the cases p, (p, 0), for all rational primes <= 32717.

Examples

			The array N1(a, b) begins:
a \ b  0   1   2   3   4   5   6   7   8   9   10 ...
-----------------------------------------------------
0:     0  -1  -4  -9 -16 -25 -36 -49 -64 -81 -100 ...
1:     1   1  -1  -5 -11 -19 -29 -41 -55 -71  -89 ...
2:     4   5   4   1  -4 -11 -20 -31 -44 -59  -76 ...
3:     9  11  11   9   5  -1  -9 -19 -31 -45  -61 ...
4:    16  19  20  19  16  11   4  -5 -16 -29  -44 ...
5:    25  29  31  31  29  25  19  11   1 -11  -25 ...
6:    36  41  44  45  44  41  36  29  20   9   -4 ...
7:    49  55  59  61  61  59  55  49  41  31   19 ...
8:    64  71  76  79  80  79  76  71  64  55   44 ...
9:    81  89  95  99 101 101  99  95  89  81   71 ...
10:  100 109 116 121 124 125 124 121 116 109  100 ...
...
-----------------------------------------------------
The Triangle T(n, k) begins:
n \ k  0  1  2  3   4   5   6   7   8   9   10 ...
0:     0
1:     1 -1
2:     4  1 -4
3:     9  5 -1 -9
4:    16 11  4 -5 -16
5:    25 19 11  1 -11 -25
6:    36 29 20  9  -4 -19 -36
7:    49 41 31 19   5 -11 -29 -49
8:    64 55 44 31  16  -1 -20 -41 -64
9:    81 71 59 45  29  11  -9 -31 -55 -81
10:  100 89 76 61  44  25   4 -19 -44 -71 -100
...
------------------------------------------------
Units from norm N(a, b) = N1(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), (1, 2), (2, 3), (3, 5), (5, 8), ...; cases + or - phi^n, n >= 0.
Some primes im Q(phi) from |N1(a, b)| = q, with q a prime in Q:
a = 1:  (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 9), (1, 10), ...
a = 2:  (2, 1), (2, 5), (2, 7), (2, 9), ...
a = 3:  (3, 1), (3, 2), (3, 4), (3, 7), (3, 8), (3, 10), ...
a = 4:  (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), ...
a = 5:  (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (5, 7), (5, 9), ...
a = 6:  (6, 1), (6, 5), (6, 7), ...
a = 7:  (7, 2), (7, 3), (7, 4), (7, 5), (7, 8), (7, 9), (7, 10), ...
a = 8:  (8, 1), (8, 3), (8, 5), (8, 7), ...
a = 9:  (9, 1), (9, 4), (9, 5), (9, 8), (9, 10), ...
a = 10: (10, 1), (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. A003631, A281385, A132111, A281385, A344686.

Formula

Array N1(a, b) = a^2 + a*b - b^2, for a >= 0 and b >= 0.

Triangle T(n, k) = N1(n-k, k) = n^2 - 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., g.f. of the triangle: G(x, y) = x*(1 - y + (1 + y - y^2)*x - 2*y*(2 - y)*x^2 + y^2*x^3)/((1 - x*y)^3*(1 - x)^3) (compare with the g.f. in A281385).