OFFSET
0,2
COMMENTS
The Fringe indices reference the double indexed Zernike polynomials with a single ordinal. Although the set of Fringe indices is limited in practical applications, the mapping covers the entire set of polynomials.
REFERENCES
Jim Schwiegerling, "Optical Specification, Fabrication, and Testing", SPIE, 2014, p. 90.
LINKS
Gerhard Ramsebner, Table of n, a(n) for n = 0..10000
Gerhard Ramsebner, animated SVG
Gerhard Ramsebner, PDF
Wikipedia, Fringe / University of Arizona indices
FORMULA
T(n,k) = (1 + (n + abs(m))/2)^2 - 2*abs(m) + [m < 0], where m = -n+2*k and [] is the Iverson bracket.
EXAMPLE
(0,0) 1
(1,-1) (1,1) 3 2
(2,-2) (2,0) (2,2) 6 4 5
(3,-3) (3,-1) (3,1) (3,3) 11 8 7 10
(4,-4) (4,-2) (4,0) (4,2) (4,4) 18 13 9 12 17
PROG
(PARI) T(n, k)=my(m=-n+2*k); (1 + (n + abs(m))/2)^2 - 2*abs(m) + (m < 0) \\ Andrew Howroyd, Aug 27 2024
CROSSREFS
KEYWORD
AUTHOR
Gerhard Ramsebner, Aug 25 2024
STATUS
approved