A145009 Array read by antidiagonals: array of odd integers found in |A144912| with axes b = {4, 6, 8, ...} and n = {b^2, b^4, b^6, ...}.

7, 13, 13, 19, 23, 19, 25, 33, 33, 25, 31, 43, 47, 43, 31, 37, 53, 61, 61, 53, 37, 43, 63, 75, 79, 75, 63, 43, 49, 73, 89, 97, 97, 89, 73, 49, 55, 83, 103, 115, 119, 115, 103, 83, 55, 61, 93, 117, 133, 141, 141, 133, 117, 93, 61

Offset: 0

Author

Reikku Kulon, Sep 28 2008

Keywords

• nonn
• tabl
• easy

Comments

The complete array can be defined as 6(x + y) + 4xy + 7.

Values along the edges are given by 6x + 7 and thus include the larger member of every twin prime pair except 5. The smaller member, 6x + 5, is adjacent in |A144912|.

Taking the origin to be z = 1, the main diagonal is given by 4z^2 + 4z - 1 (A073577).

Sums along antidiagonals are given by z(2z^2 + 12z + 7) / 3.

From Reikku Kulon, Sep 29 2008: (Start)

Any entry in the triangle can be produced from the two terms diagonally above or below and the edges can be found by taking the odd numbers as the "missing" values, starting from 1. If the terms are denoted:

.. a0 .. ...

a1 .. a2 ...

.. a3 .. ...

then:

a0 = (a1 + a2)/2 - 4*(a1 + a2 + 4)/(a2 - a1);

a3 = (a1 + a2)/2 + 4*(a1 + a2 + 4)/(a2 - a1). [Corrected by Jinyuan Wang, Jul 29 2020]

(End)

Examples

			Array A(n,k) begins:
7,  13, 19, 25,  31,  37,  43,  ...
13, 23, 33, 43,  53,  63,  73,  ...
19, 33, 47, 61,  75,  89,  103, ...
25, 43, 61, 79,  97,  115, 133, ...
31, 53, 75, 97,  119, 141, 163, ...
37, 63, 89, 115, 141, 167, 193, ...
...
		

Crossrefs

Cf. A000040, A006512, A073577, A144912.

Formula

A(n, k) = |A144912(2*n+4, (2*n+4)^(2*k+2))| = 6*(n+k) + 4*n*k + 7.