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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.

%I A176988 #28 Apr 02 2025 04:38:54
%S A176988 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 A176988 26,28,35,33,31,29,30,32,34,36,45,43,41,39,37,38,40,42,44,55,53,51,49,
%U A176988 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 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.
%C A176988 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 A176988    (0,0)
%C A176988    (1,-1) (1,1)
%C A176988    (2,-2) (2,0) (2,2)
%C A176988    (3,-3) (3,-1) (3,1) (3,3)
%C A176988    (4,-4) (4,-2) (4,0) (4,2) (4,4)
%C A176988    (5,-5) (5,-3) (5,-1) (5,1) (5,3) (5,5)
%C A176988    (6,-6) (6,-4) (6,-2) (6,0) (6,2) (6,4) (6,6)
%C A176988    (7,-7) (7,-5) (7,-3) (7,-1) (7,1) (7,3) (7,5) (7,7)
%C A176988 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 A176988    1,
%C A176988    3, 2,
%C A176988    5, 4, 6,
%C A176988    9, 7, 8,10,
%C A176988   15,13,11,12,14,
%C A176988   21,19,17,16,18,20,
%C A176988   27,25,23,22,24,26,28,
%C A176988   35,33,31,29,30,32,34,36,
%C A176988 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 A176988 N. Chetty and 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 A176988 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 A176988 Gerhard Ramsebner, <a href="/A176988/a176988.svg">Nollindex of the Zernike polynomials (animated SVG)</a>
%H A176988 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 A176988 Wikipedia, <a href="http://en.wikipedia.org/wiki/Zernike_polynomials">Zernike Polynomials</a>
%H A176988 <a href="/index/Per#IntegerPermutation">Index to sequences related to the permutation of the positive integers</a>
%F A176988 T(n,k) = n*(n+1)/2 + abs(m) + h where 0<=k<=n, j=k+n*(n+1)/2, m=2*j-n*(n+2) and h=1 if mod(n,4)<=1 and m<=0 or mod(n,4)>1 and m>=0 otherwise h=0. - _Gerhard Ramsebner_, Nov 10 2024
%p A176988 Noll := proc(n,m)
%p A176988     n*(n+1)/2+abs(m) ;
%p A176988     if m>=0 and modp(n,4) in {2,3} then
%p A176988         %+1 ;
%p A176988     elif m<=0 and modp(n,4) in {0,1} then
%p A176988         %+1 ;
%p A176988     else
%p A176988         % ;
%p A176988     end if;
%p A176988 end proc:
%p A176988 A176988 := proc(n,k)
%p A176988     Noll(n,-n+2*k) ;
%p A176988 end proc:
%p A176988 seq(seq(A176988(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Mar 27 2025
%o A176988 (PARI) A176988(n, k) = my(j=k+n*(n+1)/2, m=2*j-n*(n+2)); (n*(n+1)/2 + abs(m) + ((n%4<=1 && m<=0) || (n%4>1 && m>=0)) );
%o A176988 row(n) = vector(n+1, k, A176988(n, k-1)); \\ _Gerhard Ramsebner_, Nov 10 2024
%K A176988 nonn,easy,tabl
%O A176988 0,2
%A A176988 _R. J. Mathar_, Dec 08 2010