A375779 Noll index series of Zernike polynomials converted to ANSI index.

0, 2, 1, 4, 3, 5, 7, 8, 6, 9, 12, 13, 11, 14, 10, 18, 17, 19, 16, 20, 15, 24, 23, 25, 22, 26, 21, 27, 31, 32, 30, 33, 29, 34, 28, 35, 40, 41, 39, 42, 38, 43, 37, 44, 36, 50, 49, 51, 48, 52, 47, 53, 46, 54, 45, 60, 59, 61, 58, 62, 57, 63, 56, 64, 55, 65, 71, 72, 70, 73, 69, 74, 68, 75, 67, 76, 66, 77

Offset: 1

Author

Gerhard Ramsebner, Aug 27 2024

Keywords

• nonn
• tabl
• easy

Comments

ANSI indices of Zernike polynomials sorted by Noll index.

Examples

			Noll indices     ANSI indices
 1                0
 3 2              1 2
 5 4 6            3 4 5
 9 7 8 10         6 7 8 9
 15 13 11 12 14   10 11 12 13 14
 ...              ...
		

Links

• Gerhard Ramsebner, Table of n, a(n) for n = 1..10000
• Robert J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am. 1976, 66, 207-211.
• Gerhard Ramsebner, Noll index of Zernike polynomials (animated SVG)
• Gerhard Ramsebner, PDF
• Wikipedia, Noll's sequential indices

Crossrefs

Cf. A176988.

Programs

Maple

A375779 := proc(j::integer)
    n := floor((sqrt(8*(j-1)+1)-1)/2) ;
    m := (-1)^j*(modp(n,2)+2*floor((j-n*(n+1)/2-1+modp(n+1,2))/2)) ;
    (n*(n+2)+m)/2 ;
end proc:
seq(A375779(j),j=1..40) ; # R. J. Mathar, Mar 27 2025

PARI

for(j=1, 28, my(n=floor((sqrt(8*(j-1)+1)-1)/2)); my(m=(-1)^j*(n%2+2*floor((j-n*(n+1)/2-1+(n+1)%2)/2))); print(j,",",(n*(n+2)+m)/2))

Formula

a(j) = (n(n+2)+m)/2 where n=floor( (sqrt(8*(j-1)+1)-1)/2 ) =A003056(j-1) and m = (-1)^j *( mod(n,2)+2*floor((j-n*(n+1)/2-1+mod(n+1,2))/2) ).

Quasi-inverse: A176988(1+a(n)) = n assuming offset 1 in A176988 and serialized format. - R. J. Mathar, Mar 26 2025