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 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0,4`
	COMMENTS	`The general array N(a, b) gives the norms of the integers alpha = a1 + bphi, 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) := alphaalpha' = a^2 + ab - b^2, with alpha' = (a+b)1 - bphi. (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 - ab - b^2, for a >= 0 and b >= 0.` `Triangle T(n, k) = N2(n-k, k) = N(n-k, -k) = n^2 - 3nk + 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 - 1y^2)x - y(2 - 4y)x^2 - y^2x^3)/((1 - xy)^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	`Cf. A281385, A344685.` `Sequence in context: A344947 A079185 A133819 * A274092 A349039 A021245` `Adjacent sequences: A344683 A344684 A344685 * A344687 A344688 A344689`
	KEYWORD	`sign,tabl,easy`
	AUTHOR	`Wolfdieter Lang, Jun 17 2021`
	STATUS	`approved`