This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342354 #31 Mar 09 2021 10:39:53
%S A342354 1,3,7,9,5,17,19,11,15,31,33,21,13,29,49,51,35,23,27,47,71,73,53,37,
%T A342354 25,45,69,97,99,75,55,39,43,67,95,127,129,101,77,57,41,65,93,125,161,
%U A342354 163,131,103,79,59,63,91,123,159,199,201,165,133,105,81,61,89,121,157,197,241,243,203,167,135,107,83,87,119,155,195,239,287
%N A342354 M(n,k) = 2*n^2 + 2*k + 1 for 0 <= k <= n and M(n,k) = 2*k^2 + 4*k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).
%C A342354 This is a square array defined by _J. M. Bergot_ in A005917 (originally by mistake in A047926). Here is the edited description of the array by this contributor.
%C A342354 Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals A005917(n+1).
%F A342354 O.g.f. for rectangular M: (x^4*y^4 + 4*x^3*y^4 + 3*x^4*y^2 - 18*x^3*y^3 - x^2*y^4 + 8*x^3*y^2 + 4*x^2*y^3 - 10*x^3*y + 10*x^2*y^2 - 2*x*y^3 + 8*x^2*y + 4*x*y^2 + 3*x^2 - 18*x*y - y^2 + 4*y + 1)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2).
%F A342354 O.g.f. for triangular T: (x^8*y^4 + 4*x^7*y^4 - x^6*y^4 - 18*x^6*y^3 + 3*x^6*y^2 + 4*x^5*y^3 + 8*x^5*y^2 - 2*x^4*y^3 + 10*x^4*y^2 - 10*x^4*y + 4*x^3*y^2 + 8*x^3*y - x^2*y^2 - 18*x^2*y + 3*x^2 + 4*x*y + 1)/((1 - x)^3*(1 - x*y)^3*(1 - x^2*y)^2).
%e A342354 Square array M(n,k) (n, k >= 0) begins:
%e A342354 1, 7, 17, 31, 49, 71, 97, 127, ...
%e A342354 3, 5, 15, 29, 47, 69, 95, 125, ...
%e A342354 9, 11, 13, 27, 45, 67, 93, 123, ...
%e A342354 19, 21, 23, 25, 43, 65, 91, 121, ...
%e A342354 33, 35, 37, 39, 41, 63, 89, 119, ...
%e A342354 51, 53, 55, 57, 59, 61, 87, 117, ...
%e A342354 73, 75, 77, 79, 81, 83, 85, 115, ...
%e A342354 ...
%e A342354 The triangular array T(n,k) = M(n-k,k) (with rows n >= 0 and columns k = 0..n) is obtained by reading array M by ascending antidiagonals:
%e A342354 1;
%e A342354 3, 7;
%e A342354 9, 5, 17;
%e A342354 19, 11, 15, 31;
%e A342354 33, 21, 13, 29, 49;
%e A342354 51, 35, 23, 27, 47, 71;
%e A342354 73, 53, 37, 25, 45, 69, 97;
%e A342354 99, 75, 55, 39, 43, 67, 95, 127;
%e A342354 ...
%o A342354 (PARI) tabl(nn) = {my(M=matrix(nn+1,nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n,k]=if(k == n, 2*n^2-2*n+1, if(k < n, 2*n^2-4*n+2*k+1, 2*k^2-2*n+1)))); M}
%Y A342354 Cf. A001844, A005917, A047926, A056220, A058331.
%Y A342354 Antidiagonal sums are in A342362.
%K A342354 nonn,tabl
%O A342354 0,2
%A A342354 _Petros Hadjicostas_, Mar 08 2021