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

1,2

Comments

ANSI indices of Zernike polynomials sorted by Noll index.

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

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

Example

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
 ...              ...

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

Prog

(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))

Crossrefs

Cf. A176988.

Sequence in context: A324755 A259019 A387325 * A262663 A375469 A075348

Adjacent sequences: A375776 A375777 A375778 * A375780 A375781 A375782

Keyword

nonn,tabl,easy

Author

Gerhard Ramsebner, Aug 27 2024

Status

approved