A132111 -id:A132111 - cp's OEIS frontend

A352242 Regular triangle T(n,k) = (n-k)*(n^3-k^3) for n>=2 and 1 <= k <= n-1, read by rows.

7, 52, 19, 189, 112, 37, 496, 351, 196, 61, 1075, 832, 567, 304, 91, 2052, 1675, 1264, 837, 436, 127, 3577, 3024, 2425, 1792, 1161, 592, 169, 5824, 5047, 4212, 3325, 2416, 1539, 772, 217, 8991, 7936, 6811, 5616, 4375, 3136, 1971, 976, 271, 13300, 11907, 10432, 8869, 7236, 5575, 3952, 2457, 1204, 331

Offset: 2

Michel Marcus, Mar 09 2022

			Triangle begins:
     7;
    52,   19;
   189,  112,   37;
   496,  351,  196,  61;
  1075,  832,  567, 304,  91;
  2052, 1675, 1264, 837, 436, 127;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows n = 2..150, flattened)

Cf. A003215 (right diagonal), A138849 (left border).
Cf. A352243, A352244.
Cf. A055461, A132111.

Table[(n - k) (n^3 - k^3), {n, 2, 11}, {k, n - 1}] // Flatten (* Michael De Vlieger, Mar 09 2022 *)

row(n) = vector(n-1, k, (n-k)*(n^3-k^3));

T(n,k) = A055461(n,k)*A132111(n,k). - R. J. Mathar, Feb 10 2025

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.

Original entry on oeis.org

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

Wolfdieter Lang, Jun 17 2021

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.

			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) ...
...

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.

Cf. A003631, A281385, A132111, A281385, A344686.

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).

cp's OEIS Frontend

A352242 Regular triangle T(n,k) = (n-k)*(n^3-k^3) for n>=2 and 1 <= k <= n-1, read by rows.

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