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A061579 Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.

%I A061579 #50 Jun 06 2024 23:24:33
%S A061579 0,2,1,5,4,3,9,8,7,6,14,13,12,11,10,20,19,18,17,16,15,27,26,25,24,23,
%T A061579 22,21,35,34,33,32,31,30,29,28,44,43,42,41,40,39,38,37,36,54,53,52,51,
%U A061579 50,49,48,47,46,45,65,64,63,62,61,60,59,58,57,56,55,77,76,75,74,73,72,71,70,69,68,67,66
%N A061579 Reverse one number (0), then two numbers (2,1), then three (5,4,3), then four (9,8,7,6), etc.
%C A061579 A self-inverse permutation of the nonnegative numbers.
%C A061579 a(n) is the smallest nonnegative integer not yet in the sequence such that n + a(n) is one less than a square. - _Franklin T. Adams-Watters_, Apr 06 2009
%C A061579 From _Michel Marcus_, Mar 01 2021: (Start)
%C A061579 Array T(n,k) = (n+k)^2/2 + (n+3*k)/2 for n,k >= 0 read by descending antidiagonals.
%C A061579 Array T(n,k) = (n+k)^2/2 + (3*n+k)/2 for n,k >= 0 read by ascending antidiagonals. (End)
%H A061579 Harry J. Smith, <a href="/A061579/b061579.txt">Table of n, a(n) for n = 0..1000</a>
%H A061579 Madeline Brandt and Kåre Schou Gjaldbæk, <a href="https://arxiv.org/abs/2102.13578">Classification of Quadratic Packing Polynomials on Sectors of R^2</a>, arXiv:2102.13578 [math.NT], 2021. See Figure 9 p. 17.
%H A061579 Gennady Eremin, <a href="https://arxiv.org/abs/2405.16143">Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant</a>, arXiv:2405.16143 [math.CO], 2024.
%H A061579 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A061579 a(n) = floor(sqrt(2n+1)-1/2)*floor(sqrt(2n+1)+3/2) - n = A005563(A003056(n)) - n.
%F A061579 Row (or antidiagonal) n = 0, 1, 2, ... contains the integers from A000217(n) to A000217(n+1)-1 in reverse order (for diagonals, "reversed" with respect to the canonical "falling" order, cf. A001477/table). - _M. F. Hasler_, Nov 09 2021
%F A061579 From _Alois P. Heinz_, Feb 10 2023: (Start)
%F A061579 T(n,k) = n*(n+3)/2 - k.
%F A061579 Sum_{k=0..n} k * T(n,k) = A002419(n).
%F A061579 Sum_{k=0..n} k^2 * T(n,k) = A119771(n).
%F A061579 Sum_{k=0..n} (-1)^k * T(n,k) = A226725(n). (End)
%e A061579 Read as a triangle, the sequence is:
%e A061579     0
%e A061579     2   1
%e A061579     5   4   3
%e A061579     9   8   7   6
%e A061579    14  13  12  11  10
%e A061579   (...)
%e A061579 As an infinite square matrix (cf. the "table" link, 2nd paragraph) it reads:
%e A061579     0    2    5    9   14   20   ...
%e A061579     1    4    8   13   19   22   ...
%e A061579     3    7   12   18   23   30   ...
%e A061579     6   11   17   24   31   39   ...
%e A061579   (...)
%p A061579 T:= (n,k)-> n*(n+3)/2-k:
%p A061579 seq(seq(T(n,k), k=0..n), n=0..12);  # _Alois P. Heinz_, Feb 10 2023
%t A061579 Module[{nn=20},Reverse/@TakeList[Range[0,(nn(nn+1))/2],Range[nn]]]// Flatten (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Jul 06 2018 *)
%o A061579 (PARI) A061579_row(n)=vector(n+=1, j, n*(n+1)\2-j)
%o A061579 A061579_upto(n)=concat([A061579_row(r)|r<-[0..sqrtint(2*n)]]) \\ yields approximately n terms: actual number differs by less than +- sqrt(n). - _M. F. Hasler_, Nov 09 2021
%o A061579 (Python)
%o A061579 from math import isqrt
%o A061579 def A061579(n): return (r:=isqrt((n<<3)+1)-1>>1)*(r+2)-n # _Chai Wah Wu_, Feb 10 2023
%Y A061579 Fixed points are A046092.
%Y A061579 Row sums give A027480.
%Y A061579 Each reversal involves the numbers from A000217 through to A000096.
%Y A061579 Cf. A038722. Transpose of A001477.
%Y A061579 Cf. A002419, A119771, A226725.
%K A061579 nonn,tabl
%O A061579 0,2
%A A061579 _Henry Bottomley_, May 21 2001