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A375510
Fringe indices of Zernike polynomials.
1
1, 3, 2, 6, 4, 5, 11, 8, 7, 10, 18, 13, 9, 12, 17, 27, 20, 15, 14, 19, 26, 38, 29, 22, 16, 21, 28, 37, 51, 40, 31, 24, 23, 30, 39, 50, 66, 53, 42, 33, 25, 32, 41, 52, 65, 83, 68, 55, 44, 35, 34, 43, 54, 67, 82, 102, 85, 70, 57, 46, 36, 45, 56, 69, 84, 101, 123, 104, 87, 72, 59, 48, 47, 58, 71, 86, 103, 122, 146, 125, 106, 89, 74, 61, 49, 60, 73
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, animated SVG
Gerhard Ramsebner, PDF
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
Cf. A176988.
Sequence in context: A274315 A194059 A191427 * A191428 A191733 A191444
KEYWORD
nonn,easy,tabl
AUTHOR
Gerhard Ramsebner, Aug 25 2024
STATUS
approved