A054082 Permutation of N: a(1)=2, a(2)=1 and for each k >= 2, let p(k)=least natural number not already an a(i), q(k)=p(k)+k-1, a(2k-1)=q(k), a(2k)=p(k).
2, 1, 4, 3, 7, 5, 9, 6, 12, 8, 15, 10, 17, 11, 20, 13, 22, 14, 25, 16, 28, 18, 30, 19, 33, 21, 36, 23, 38, 24, 41, 26, 43, 27, 46, 29, 49, 31, 51, 32, 54, 34, 56, 35, 59, 37, 62, 39, 64, 40, 67, 42, 70, 44, 72, 45, 75, 47, 77, 48, 80, 50, 83
Offset: 1
Programs
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Mathematica
a[n_] := If[OddQ[n], Floor[((n+1)/2 - 1) GoldenRatio] + (n+1)/2 + 1, Floor[(n/2 - 1) GoldenRatio] + 2]; a[2] = 1; Array[a, 100] (* Jean-François Alcover, Apr 01 2020 *)