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

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

Offset: 0

Author

Henry Bottomley, May 21 2001

Keywords

• nonn
• tabl

Comments

A self-inverse permutation of the nonnegative numbers.

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

From Michel Marcus, Mar 01 2021: (Start)

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

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

Examples

			Read as a triangle, the sequence is:
    0
    2   1
    5   4   3
    9   8   7   6
   14  13  12  11  10
  (...)
As an infinite square matrix (cf. the "table" link, 2nd paragraph) it reads:
    0    2    5    9   14   20   ...
    1    4    8   13   19   22   ...
    3    7   12   18   23   30   ...
    6   11   17   24   31   39   ...
  (...)
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Madeline Brandt and Kåre Schou Gjaldbæk, Classification of Quadratic Packing Polynomials on Sectors of R^2, arXiv:2102.13578 [math.NT], 2021. See Figure 9 p. 17.
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Fixed points are A046092.

Row sums give A027480.

Each reversal involves the numbers from A000217 through to A000096.

Cf. A038722. Transpose of A001477.

Cf. A002419, A119771, A226725.

Programs

Maple

T:= (n,k)-> n*(n+3)/2-k:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Feb 10 2023

Mathematica

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 *)

PARI

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

Python

from math import isqrt
def A061579(n): return (r:=isqrt((n<<3)+1)-1>>1)*(r+2)-n # Chai Wah Wu, Feb 10 2023

Formula

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

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

From Alois P. Heinz, Feb 10 2023: (Start)

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

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

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

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