A344686 - OEIS

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

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OFFSET

0,4

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.

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.

LINKS

Table of n, a(n) for n=0..65.

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).

EXAMPLE

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), ...

CROSSREFS