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) and m=(-1)^j*(mod(n,2)+2*floor((j-n*(n+1)/2-1+mod(n+1,2))/2))
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
... ...
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
KEYWORD
AUTHOR
Gerhard Ramsebner, Aug 27 2024
STATUS
approved