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 A385406 #10 Jun 28 2025 12:53:00
%S A385406 1,3,2,5,4,6,9,8,10,7,13,12,14,11,15,19,18,20,17,21,16,25,24,26,23,27,
%T A385406 22,28,33,32,34,31,35,30,36,29,41,40,42,39,43,38,44,37,45,51,50,52,49,
%U A385406 53,48,54,47,55,46,61,60,62,59,63,58,64,57,65,56,66,73,72,74,71,75,70,76,69,77,68,78,67
%N A385406 Triangle read by rows: T(n, k) = n*(n+1)/2 - floor((n-1)/2) - (-1)^k * floor(k/2).
%C A385406 This triangle seen as a sequence yields a permutation of the natural numbers (A000027).
%H A385406 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A385406 T(n, k) = T(n, k-1) - (-1)^k * (k-1) for 1 < k <= n with initial values T(n, 1) = n*(n+1)/2 - floor((n-1)/2) for n >= 1.
%F A385406 T(n, n) = n*(n+1)/2 + (1-n) * (1 - n mod 2) = A128918(n).
%F A385406 T(2*n-1, n) = 2*n^2 - 2*n + 1 - (-1)^n * floor(n/2) = A213399(n-1).
%e A385406 Triangle T(n, k) for 1 <= k <= n starts:
%e A385406 n \k : 1 2 3 4 5 6 7 8 9 10 11 12 13
%e A385406 ==========================================================
%e A385406 1 : 1
%e A385406 2 : 3 2
%e A385406 3 : 5 4 6
%e A385406 4 : 9 8 10 7
%e A385406 5 : 13 12 14 11 15
%e A385406 6 : 19 18 20 17 21 16
%e A385406 7 : 25 24 26 23 27 22 28
%e A385406 8 : 33 32 34 31 35 30 36 29
%e A385406 9 : 41 40 42 39 43 38 44 37 45
%e A385406 10 : 51 50 52 49 53 48 54 47 55 46
%e A385406 11 : 61 60 62 59 63 58 64 57 65 56 66
%e A385406 12 : 73 72 74 71 75 70 76 69 77 68 78 67
%e A385406 13 : 85 84 86 83 87 82 88 81 89 80 90 79 91
%e A385406 etc.
%t A385406 T[n_, k_] := n*(n+1)/2 - Floor[(n-1)/2] - (-1)^k*Floor[k/2]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jun 28 2025 *)
%o A385406 (PARI) T(n, k) = n*(n+1)/2 - floor((n-1)/2) - (-1)^k * floor(k/2)
%Y A385406 Cf. A080827 (column 1), A128918 (main diagonal), A006003 (row sums), A213399.
%K A385406 nonn,easy,tabl
%O A385406 1,2
%A A385406 _Werner Schulte_, Jun 27 2025