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A066182 Permutation of the integers with cycle form {1}, {3, 2}, {6, 5, 4}, {10, 9, 8, 7}, ...

%I A066182 #36 Jun 09 2025 14:40:49
%S A066182 1,3,2,6,4,5,10,7,8,9,15,11,12,13,14,21,16,17,18,19,20,28,22,23,24,25,
%T A066182 26,27,36,29,30,31,32,33,34,35,45,37,38,39,40,41,42,43,44,55,46,47,48,
%U A066182 49,50,51,52,53,54,66,56,57,58,59,60,61,62,63,64,65,78,67,68,69,70,71
%N A066182 Permutation of the integers with cycle form {1}, {3, 2}, {6, 5, 4}, {10, 9, 8, 7}, ...
%C A066182 Arrange natural numbers 1,2,3,4,5,... as a triangle like A000027, then rotate each row of triangle one step right. - _Antti Karttunen_, May 07 2002
%C A066182 As a rectangular array, a(n) is the natural interspersion of the sequence of triangular numbers; see A192872. [_Clark Kimberling_, Aug 12 2011]
%H A066182 Matthew House, <a href="/A066182/b066182.txt">Table of n, a(n) for n = 1..45150</a>
%H A066182 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A066182 a(n) = -1+n+binomial(A002024(n)+1,2)-binomial(A002024(n-1)+1,2) where A002024(n) is round(sqrt(2*n)). - _Brian Tenneson_, Feb 03 2017
%e A066182 Northwest corner, when sequence is formatted as the natural interspersion of the sequence (1,3,6,10,15,...) of triangular numbers:
%e A066182   1...3...6...10...15
%e A066182   2...4...7...11...16
%e A066182   5...8...12..17...23
%e A066182   9...13..18..24...31     [ _Clark Kimberling_, Aug 12 2011 ]
%t A066182 FromCycles[Table[n(n-1)/2+Range[n, 1, -1], {n, 13}]]
%o A066182 (Python)
%o A066182 from math import isqrt, comb
%o A066182 def A066182(n): return -1+n+comb(isqrt(n<<3)+3>>1,2)-comb(isqrt(n-1<<3)+3>>1,2) # _Chai Wah Wu_, Jun 09 2025
%Y A066182 Inverse permutation: A066181.
%Y A066182 Cf. A000027, A192872.
%K A066182 easy,nonn,tabl
%O A066182 1,2
%A A066182 _Wouter Meeussen_, Dec 15 2001