A377850 Noll index series of Zernike polynomials converted to Fringe index.

1, 2, 3, 4, 6, 5, 8, 7, 11, 10, 9, 12, 13, 17, 18, 14, 15, 19, 20, 26, 27, 16, 22, 21, 29, 28, 38, 37, 24, 23, 31, 30, 40, 39, 51, 50, 25, 32, 33, 41, 42, 52, 53, 65, 66, 34, 35, 43, 44, 54, 55, 67, 68, 82, 83, 36, 46, 45, 57, 56, 70, 69, 85, 84, 102, 101, 48, 47, 59, 58, 72, 71, 87, 86

Offset: 1

Author

Gerhard Ramsebner, Nov 09 2024

Keywords

• nonn
• easy

Comments

Fringe indices of Zernike polynomials sorted by Noll index.

Examples

			  Noll indices      Fringe indices
   1                 1
   3  2              3  2
   5  4  6           6  4 5
   9  7  8 10       11  8 7 10
  15 13 11 12 14    18 13 9 12 17
  ...               ...
		

Links

• Gerhard Ramsebner, Table of n, a(n) for n = 1..10000
• Gerhard Ramsebner, Noll index series of Zernike polynomials converted to Fringe index
• Gerhard Ramsebner, Noll index (animated SVG)
• Gerhard Ramsebner, Fringe index (animated SVG)
• Gerhard Ramsebner, Plot

Crossrefs

Cf. A176988, A375510.

Programs

PARI

A377850(j) = my(n=floor( (sqrt(8*(j-1)+1)-1)/2 ), m=(-1)^j*(n%2+2*floor((j-n*(n+1)/2-1+(n+1)%2)/2))); (1+(n+abs(m))/2)^2 -2*abs(m)+(m<0);

Formula

a(j) = (1+(n+abs(m))/2)^2-2*abs(m)+[m<0] where n=floor((sqrt(8*(j-1)+1)-1)/2), m=(-1)^j*(mod(n,2)+2*floor((j-n*(n+1)/2-1+mod(n+1,2))/2)) and [] is the Iverson bracket.

a(A176988(j)) = A375510(j) assuming offset 1 in all 3 sequences and serialized versions.