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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 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|.

REFERENCES

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

Thomas Risse, Least Square Approximation with Zernike Polynomials Using SAGE, http://www.weblearn.hs-bremen.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

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Last modified December 7 08:21 EST 2016. Contains 278849 sequences.