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
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.
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.
Comments