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 A367269 #32 Jan 10 2024 04:54:44
%S A367269 1,4,3,6,5,2,13,12,9,8,15,14,11,10,7,26,25,22,21,18,17,28,27,24,23,20,
%T A367269 19,16,43,42,39,38,35,34,31,30,45,44,41,40,37,36,33,32,29,64,63,60,59,
%U A367269 56,55,52,51,48,47,66,65,62,61,58,57,54,53,50,49,46,89,88,85,84,81,80,77,76,73,72,69,68
%N A367269 Triangle T(n, k) read by rows and based on A042948 yields a permutation of the natural numbers.
%C A367269 Compare this triangle to A364390.
%F A367269 T(n, k) = (n+1) * (n+2) / 2 + n * (n mod 2) - 2 * k + (k mod 2) for 0 <= k <= n.
%F A367269 T(n, k) = T(n, 0) + A042948(k) for 0 <= k <= n.
%F A367269 T(n, 0) = (n+1) * (n+2) / 2 + n * (n mod 2) for n >= 0.
%F A367269 T(n, n) = (n^2 - n + 2) / 2 + (n+1) * (n mod 2) for n >= 0.
%F A367269 T(2*n, n) = 2 * n^2 + n + 1 + (n mod 2) for n >= 0.
%F A367269 T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-1) for 0 < k < n.
%F A367269 Row sums: A006003(n+1) - 2 * (-1)^n * (floor((n+1)/2))^2 for n >= 0.
%F A367269 G.f. of column k = 0: F(t, 0) = Sum_{n>=0} T(n, 0) * t^n = (1 + 3*t + t^3 - t^4) / ((1-t)^3 * (1+t)^2).
%F A367269 G.f.: F(t, x) = Sum_{n>=0, k=0..n} T(n, k) * x^k * t^n = (F(t, 0) - x * F(x*t, 0)) / (1-x) - 2*x*t / ((1-t) * (1-x*t)^2) + x*t / ((1-t) * (1-x^2*t^2)).
%F A367269 Alt. row sums: (n^(2 - n mod 2) + 2 - n mod 2) / 2 for n >= 0.
%e A367269 Triangle T(n, k) for 0 <= k <= n starts:
%e A367269 n\k : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e A367269 =================================================================
%e A367269 0 : 1
%e A367269 1 : 4 3
%e A367269 2 : 6 5 2
%e A367269 3 : 13 12 9 8
%e A367269 4 : 15 14 11 10 7
%e A367269 5 : 26 25 22 21 18 17
%e A367269 6 : 28 27 24 23 20 19 16
%e A367269 7 : 43 42 39 38 35 34 31 30
%e A367269 8 : 45 44 41 40 37 36 33 32 29
%e A367269 9 : 64 63 60 59 56 55 52 51 48 47
%e A367269 10 : 66 65 62 61 58 57 54 53 50 49 46
%e A367269 11 : 89 88 85 84 81 80 77 76 73 72 69 68
%e A367269 12 : 91 90 87 86 83 82 79 78 75 74 71 70 67
%e A367269 13 : 118 117 114 113 110 109 106 105 102 101 98 97 94 93
%e A367269 14 : 120 119 116 115 112 111 108 107 104 103 100 99 96 95 92
%e A367269 etc.
%t A367269 T[n_, k_]:= (n+1) * (n+2) / 2 + n * Mod[n,2] - 2 * k + Mod[k,2]; Table[T[n,k],{n,0,11},{k,0,n}]//Flatten (* _Stefano Spezia_, Dec 06 2023 *)
%o A367269 (PARI) T(n, k) = (n+1)*(n+2)/2+n*(n%2)-2*k+(k%2)
%Y A367269 Cf. A006003, A042948, A364390.
%K A367269 nonn,easy,tabl
%O A367269 0,2
%A A367269 _Werner Schulte_, Dec 06 2023