

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.


0



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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



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 doubleindex (n,m) has become a defacto 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.


REFERENCES

N. Chetty, D. J. Griffith, Zernikebasis expansion of the fractional and radial Hilbert phase masks, Current Applied Physics, 15 (2015) 739747
Thomas Risse, Least Square Approximation with Zernike Polynomials Using SAGE, http://www.weblearn.hsbremen.de/risse/papers/SiP27/Zernike.pdf.


LINKS

Table of n, a(n) for n=0..119.
R. J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am 66 (1976) 207.
Wikipedia, Zernike Polynomials
Index to sequences related to the permutation of the positive integers


CROSSREFS

Sequence in context: A194837 A054068 A194870 * A194903 A194875 A194836
Adjacent sequences: A176985 A176986 A176987 * A176989 A176990 A176991


KEYWORD

nonn,easy,tabl


AUTHOR

R. J. Mathar, Dec 08 2010


STATUS

approved



