

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
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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

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



