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%I #12 May 05 2017 11:03:38
%S 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,
%T 26,28,35,33,31,29,30,32,34,36,45,43,41,39,37,38,40,42,44,55,53,51,49,
%U 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
%N Triangle read by rows, which contains Noll's indices of Zernike polynomials in row n sorted along increasing index of the azimuthal quantum number.
%C 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:
%C (0,0)
%C (1,-1) (1,1)
%C (2,-2) (2,0) (2,2)
%C (3,-3) (3,-1) (3,1) (3,3)
%C (4,-4) (4,-2) (4,0) (4,2) (4,4)
%C (5,-5) (5,-3) (5,-1) (5,1) (5,3) (5,5)
%C (6,-6) (6,-4) (6,-2) (6,0) (6,2) (6,4) (6,6)
%C (7,-7) (7,-5) (7,-3) (7,-1) (7,1) (7,3) (7,5) (7,7)
%C 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
%C 1,
%C 3,2,
%C 5,4,6,
%C 9,7,8,10,
%C 15,13,11,12,14,
%C 21,19,17,16,18,20,
%C 27,25,23,22,24,26,28,
%C 35,33,31,29,30,32,34,36,
%C 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|.
%H N. Chetty, D. J. Griffith, <a href="http://dx.doi.org/10.1016/j.cap.2015.03.017">Zernike-basis expansion of the fractional and radial Hilbert phase masks</a>, Current Applied Physics, 15 (2015) 739-747
%H R. J. Noll, <a href="http://dx.doi.org/10.1364/JOSA.66.000207">Zernike polynomials and atmospheric turbulence</a>, J. Opt. Soc. Am 66 (1976) 207.
%H Thomas Risse, <a href="http://www.weblearn.hs-bremen.de/risse/papers/SiP27/Zernike.pdf">Least Square Approximation with Zernike Polynomials Using SAGE</a>, (2011).
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Zernike_polynomials">Zernike Polynomials</a>
%H <a href="/index/Per#IntegerPermutation">Index to sequences related to the permutation of the positive integers</a>
%K nonn,easy,tabl
%O 0,2
%A _R. J. Mathar_, Dec 08 2010
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