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 A220508 #45 Mar 10 2021 08:52:45 %S A220508 0,1,3,4,2,8,9,5,7,15,16,10,6,14,24,25,17,11,13,23,35,36,26,18,12,22, %T A220508 34,48,49,37,27,19,21,33,47,63,64,50,38,28,20,32,46,62,80,81,65,51,39, %U A220508 29,31,45,61,79,99,100,82,66,52,40,30,44,60,78,98,120 %N A220508 T(n,k) = n^2 + k if k <= n, otherwise T(n,k) = k*(k + 2) - n; square array T(n,k) read by ascending antidiagonals (n >= 0, k >= 0). %C A220508 This sequence consists of 0 together with a permutation of the natural numbers. The nonnegative integers (A001477) are arranged in the successive layers from T(0,0) = 0. The n-th layer start with T(n,1) = n^2. The n-th layer is formed by the first n+1 elements of row n and the first n elements in increasing order of the column n. %C A220508 The first antidiagonal is formed by odd numbers: 1, 3. The second antidiagonal is formed by even numbers: 4, 2, 8. The third antidiagonal is formed by odd numbers: 9, 5, 7, 15. And so on. %C A220508 It appears that in the n-th layer there is at least a prime number <= g and also there is at least a prime number > g, where g is the number on the main diagonal, the n-th oblong number A002378(n), if n >= 1. %H A220508 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> %F A220508 From _Petros Hadjicostas_, Mar 10 2021: (Start) %F A220508 T(n,k) = (A342354(n,k) - 1)/2. %F A220508 O.g.f.: (x^4*y^3 + 3*x^3*y^4 + x^4*y^2 - 10*x^3*y^3 - x^2*y^4 + 3*x^3*y^2 + x^2*y^3 - 4*x^3*y + 8*x^2*y^2 + 3*x^2*y + x*y^2 + x^2 - 10*x*y - y^2 + x + 3*y)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2). (End) %e A220508 The second layer is [4, 5, 6, 7, 8] which looks like this: %e A220508 . . 8 %e A220508 . . 7, %e A220508 4, 5, 6, %e A220508 Square array T(0,0)..T(10,10) begins: %e A220508 0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120,... %e A220508 1, 2, 7, 14, 23, 34, 47, 62, 79, 98, 119,... %e A220508 4, 5, 6, 13, 22, 33, 46, 61, 78, 97, 118,... %e A220508 9, 10, 11, 12, 21, 32, 45, 60, 77, 96, 117,... %e A220508 16, 17, 18, 19, 20, 31, 44, 59, 76, 95, 118,... %e A220508 25, 26, 27, 28, 29, 30, 43, 58, 75, 94, 117,... %e A220508 36, 37, 38, 39, 40, 41, 42, 57, 74, 93, 114,... %e A220508 49, 50, 51, 52, 53, 54, 55, 56, 73, 92, 113,... %e A220508 64, 65, 66, 67, 68, 69, 70, 71, 72, 91, 112,... %e A220508 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 111,... %e A220508 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,... %e A220508 ... %Y A220508 Column 1 is A000290. Main diagonal is A002378. Column 2 is essentially A002522. Row 1 is A005563. Row 2 gives the absolute terms of A008865. %Y A220508 Cf. A000005, A000040, A002061, A002620, A014206, A014209, A027688, A028387, A028552, A060736, A220516. %K A220508 nonn,tabl %O A220508 0,3 %A A220508 _Omar E. Pol_, Feb 09 2013 %E A220508 Name edited by _Petros Hadjicostas_, Mar 10 2021