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A375779
Noll index series of Zernike polynomials converted to ANSI index.
1
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
Robert J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am. 1976, 66, 207-211.
Gerhard Ramsebner, PDF
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
Cf. A176988.
Sequence in context: A118267 A324755 A259019 * A262663 A375469 A075348
KEYWORD
nonn,tabl,easy
AUTHOR
Gerhard Ramsebner, Aug 27 2024
STATUS
approved