cp's OEIS Frontend

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, ...}.

%I A145009 #13 Jul 30 2020 00:32:30
%S A145009 7,13,13,19,23,19,25,33,33,25,31,43,47,43,31,37,53,61,61,53,37,43,63,
%T A145009 75,79,75,63,43,49,73,89,97,97,89,73,49,55,83,103,115,119,115,103,83,
%U A145009 55,61,93,117,133,141,141,133,117,93,61
%N 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, ...}.
%C A145009 The complete array can be defined as 6(x + y) + 4xy + 7.
%C A145009 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|.
%C A145009 Taking the origin to be z = 1, the main diagonal is given by 4z^2 + 4z - 1 (A073577).
%C A145009 Sums along antidiagonals are given by z(2z^2 + 12z + 7) / 3.
%C A145009 From _Reikku Kulon_, Sep 29 2008: (Start)
%C A145009 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:
%C A145009 .. a0 .. ...
%C A145009 a1 .. a2 ...
%C A145009 .. a3 .. ...
%C A145009 then:
%C A145009 a0 = (a1 + a2)/2 - 4*(a1 + a2 + 4)/(a2 - a1);
%C A145009 a3 = (a1 + a2)/2 + 4*(a1 + a2 + 4)/(a2 - a1). [Corrected by _Jinyuan Wang_, Jul 29 2020]
%C A145009 (End)
%F A145009 A(n, k) = |A144912(2*n+4, (2*n+4)^(2*k+2))| = 6*(n+k) + 4*n*k + 7.
%e A145009 Array A(n,k) begins:
%e A145009 7,  13, 19, 25,  31,  37,  43,  ...
%e A145009 13, 23, 33, 43,  53,  63,  73,  ...
%e A145009 19, 33, 47, 61,  75,  89,  103, ...
%e A145009 25, 43, 61, 79,  97,  115, 133, ...
%e A145009 31, 53, 75, 97,  119, 141, 163, ...
%e A145009 37, 63, 89, 115, 141, 167, 193, ...
%e A145009 ...
%Y A145009 Cf. A000040, A006512, A073577, A144912.
%K A145009 nonn,tabl,easy
%O A145009 0,1
%A A145009 _Reikku Kulon_, Sep 28 2008