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

0,2

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

Index to sequences related to the permutation of the positive integers

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

Adjacent sequences: A375507 A375508 A375509 * A375511 A375512 A375513

Keyword

nonn,easy,tabl

Author

Gerhard Ramsebner, Aug 25 2024

Status

approved