A176988 Triangle read by rows, which contains Noll's indices of Zernike polynomials in row n sorted along increasing index of the azimuthal quantum number.

1, 3, 2, 5, 4, 6, 9, 7, 8, 10, 15, 13, 11, 12, 14, 21, 19, 17, 16, 18, 20, 27, 25, 23, 22, 24, 26, 28, 35, 33, 31, 29, 30, 32, 34, 36, 45, 43, 41, 39, 37, 38, 40, 42, 44, 55, 53, 51, 49, 47, 46, 48, 50, 52, 54, 65, 63, 61, 59, 57, 56, 58, 60, 62, 64, 66, 77, 75, 73, 71, 69, 67, 68, 70, 72, 74, 76, 78, 91, 89, 87, 85, 83, 81, 79, 80, 82, 84, 86, 88, 90, 105, 103, 101, 99, 97, 95, 93, 92, 94, 96, 98, 100, 102, 104, 119, 117, 115, 113, 111, 109, 107, 106, 108, 110, 112, 114, 116, 118, 120

Offset

0,2

Comments

The natural arrangement of the indices n (radial index) and m (azimuthal index) of the Zernike polynomial Z(n,m) is a triangle with row index n, in each row m ranging from -n to n in steps of 2:

(0,0)

(1,-1) (1,1)

(2,-2) (2,0) (2,2)

(3,-3) (3,-1) (3,1) (3,3)

(4,-4) (4,-2) (4,0) (4,2) (4,4)

(5,-5) (5,-3) (5,-1) (5,1) (5,3) (5,5)

(6,-6) (6,-4) (6,-2) (6,0) (6,2) (6,4) (6,6)

(7,-7) (7,-5) (7,-3) (7,-1) (7,1) (7,3) (7,5) (7,7)

For uses in linear algebra related to beam optics, a standard scheme of assigning a single index j>=1 to each double-index (n,m) has become a de-facto standard, proposed by Noll. The triangle of the j at the equivalent positions reads

1,

3,2,

5,4,6,

9,7,8,10,

15,13,11,12,14,

21,19,17,16,18,20,

27,25,23,22,24,26,28,

35,33,31,29,30,32,34,36,

which defines the OEIS entries. The rule of translation is that odd j are assigned to m<0, even j to m>=0, and smaller j to smaller |m|.

Links

Table of n, a(n) for n=0..119.

N. Chetty, D. J. Griffith, Zernike-basis expansion of the fractional and radial Hilbert phase masks, Current Applied Physics, 15 (2015) 739-747

R. J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am 66 (1976) 207.

Thomas Risse, Least Square Approximation with Zernike Polynomials Using SAGE, (2011).

Wikipedia, Zernike Polynomials

Index to sequences related to the permutation of the positive integers

Crossrefs

Sequence in context: A194870 A355014 A302252 * A194903 A194875 A194836

Adjacent sequences: A176985 A176986 A176987 * A176989 A176990 A176991

Keyword

nonn,easy,tabl

Author

R. J. Mathar, Dec 08 2010

Status

approved