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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|. LINKS 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 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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