A375510 Fringe indices of Zernike polynomials.

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

Author

Gerhard Ramsebner, Aug 25 2024

Keywords

• nonn
• easy
• tabl

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.

Examples

			                    (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
		

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
• Index to sequences related to the permutation of the positive integers

Crossrefs

Cf. A176988.

Programs

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

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.