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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.

%I A344685 #23 Feb 03 2023 01:32:08
%S A344685 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,
%T A344685 -4,-19,-36,49,41,31,19,5,-11,-29,-49,64,55,44,31,16,-1,-20,-41,-64,
%U A344685 81,71,59,45,29,11,-9,-31,-55,-81,100,89,76,61,44,25,4,-19,-44,-71,-100
%N 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.
%C A344685 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.)
%C A344685 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.
%C A344685 The subtriangle N(a, b), with 0 <= b <= a, is given in A281385.
%C A344685 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).
%C A344685 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.
%D A344685 F. W. Dodd, Number theory in the quadratic field with golden section unit, Polygonal Publishing House, Passaic, NJ.
%D A344685 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, fifth edition, Clarendon Press Oxford, 2003.
%F A344685 Array N1(a, b) = a^2 + a*b - b^2, for a >= 0 and b >= 0.
%F A344685 Triangle T(n, k) = N1(n-k, k) = n^2 - n*k - k^2, for n >= 0 and k = 0, 1, ..., n.
%F A344685 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).
%e A344685 The array N1(a, b) begins:
%e A344685 a \ b  0   1   2   3   4   5   6   7   8   9   10 ...
%e A344685 -----------------------------------------------------
%e A344685 0:     0  -1  -4  -9 -16 -25 -36 -49 -64 -81 -100 ...
%e A344685 1:     1   1  -1  -5 -11 -19 -29 -41 -55 -71  -89 ...
%e A344685 2:     4   5   4   1  -4 -11 -20 -31 -44 -59  -76 ...
%e A344685 3:     9  11  11   9   5  -1  -9 -19 -31 -45  -61 ...
%e A344685 4:    16  19  20  19  16  11   4  -5 -16 -29  -44 ...
%e A344685 5:    25  29  31  31  29  25  19  11   1 -11  -25 ...
%e A344685 6:    36  41  44  45  44  41  36  29  20   9   -4 ...
%e A344685 7:    49  55  59  61  61  59  55  49  41  31   19 ...
%e A344685 8:    64  71  76  79  80  79  76  71  64  55   44 ...
%e A344685 9:    81  89  95  99 101 101  99  95  89  81   71 ...
%e A344685 10:  100 109 116 121 124 125 124 121 116 109  100 ...
%e A344685 ...
%e A344685 -----------------------------------------------------
%e A344685 The Triangle T(n, k) begins:
%e A344685 n \ k  0  1  2  3   4   5   6   7   8   9   10 ...
%e A344685 0:     0
%e A344685 1:     1 -1
%e A344685 2:     4  1 -4
%e A344685 3:     9  5 -1 -9
%e A344685 4:    16 11  4 -5 -16
%e A344685 5:    25 19 11  1 -11 -25
%e A344685 6:    36 29 20  9  -4 -19 -36
%e A344685 7:    49 41 31 19   5 -11 -29 -49
%e A344685 8:    64 55 44 31  16  -1 -20 -41 -64
%e A344685 9:    81 71 59 45  29  11  -9 -31 -55 -81
%e A344685 10:  100 89 76 61  44  25   4 -19 -44 -71 -100
%e A344685 ...
%e A344685 ------------------------------------------------
%e A344685 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.
%e A344685 Some primes im Q(phi) from |N1(a, b)| = q, with q a prime in Q:
%e A344685 a = 1:  (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 9), (1, 10), ...
%e A344685 a = 2:  (2, 1), (2, 5), (2, 7), (2, 9), ...
%e A344685 a = 3:  (3, 1), (3, 2), (3, 4), (3, 7), (3, 8), (3, 10), ...
%e A344685 a = 4:  (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), ...
%e A344685 a = 5:  (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (5, 7), (5, 9), ...
%e A344685 a = 6:  (6, 1), (6, 5), (6, 7), ...
%e A344685 a = 7:  (7, 2), (7, 3), (7, 4), (7, 5), (7, 8), (7, 9), (7, 10), ...
%e A344685 a = 8:  (8, 1), (8, 3), (8, 5), (8, 7), ...
%e A344685 a = 9:  (9, 1), (9, 4), (9, 5), (9, 8), (9, 10), ...
%e A344685 a = 10: (10, 1), (10, 9) ...
%e A344685 ...
%Y A344685 Cf. A003631, A281385, A132111, A281385, A344686.
%K A344685 sign,tabl,easy
%O A344685 0,4
%A A344685 _Wolfdieter Lang_, Jun 17 2021