A061579 - OEIS

%I #50 Jun 06 2024 23:24:33

%S 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 22,21,35,34,33,32,31,30,29,28,44,43,42,41,40,39,38,37,36,54,53,52,51,

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

%C A self-inverse permutation of the nonnegative numbers.

%C 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 From _Michel Marcus_, Mar 01 2021: (Start)

%C Array T(n,k) = (n+k)^2/2 + (n+3*k)/2 for n,k >= 0 read by descending antidiagonals.

%C Array T(n,k) = (n+k)^2/2 + (3*n+k)/2 for n,k >= 0 read by ascending antidiagonals. (End)

%H Harry J. Smith, <a href="/A061579/b061579.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) = floor(sqrt(2n+1)-1/2)*floor(sqrt(2n+1)+3/2) - n = A005563(A003056(n)) - n.

%F 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 From _Alois P. Heinz_, Feb 10 2023: (Start)

%F T(n,k) = n*(n+3)/2 - k.

%F Sum_{k=0..n} k * T(n,k) = A002419(n).

%F Sum_{k=0..n} k^2 * T(n,k) = A119771(n).

%F Sum_{k=0..n} (-1)^k * T(n,k) = A226725(n). (End)

%e Read as a triangle, the sequence is:

%e     0

%e     2   1

%e     5   4   3

%e     9   8   7   6

%e    14  13  12  11  10

%e   (...)

%e As an infinite square matrix (cf. the "table" link, 2nd paragraph) it reads:

%e     0    2    5    9   14   20   ...

%e     1    4    8   13   19   22   ...

%e     3    7   12   18   23   30   ...

%e     6   11   17   24   31   39   ...

%e   (...)

%p T:= (n,k)-> n*(n+3)/2-k:

%p seq(seq(T(n,k), k=0..n), n=0..12);  # _Alois P. Heinz_, Feb 10 2023

%t 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 (PARI) A061579_row(n)=vector(n+=1, j, n*(n+1)\2-j)

%o 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 (Python)

%o from math import isqrt

%o def A061579(n): return (r:=isqrt((n<<3)+1)-1>>1)*(r+2)-n # _Chai Wah Wu_, Feb 10 2023

%Y Fixed points are A046092.

%Y Row sums give A027480.

%Y Each reversal involves the numbers from A000217 through to A000096.

%Y Cf. A038722. Transpose of A001477.

%Y Cf. A002419, A119771, A226725.

%K nonn,tabl

%O 0,2

%A _Henry Bottomley_, May 21 2001