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A375510 Fringe indices of Zernike polynomials.

%I A375510 #46 Mar 26 2025 13:11:27
%S A375510 1,3,2,6,4,5,11,8,7,10,18,13,9,12,17,27,20,15,14,19,26,38,29,22,16,21,
%T A375510 28,37,51,40,31,24,23,30,39,50,66,53,42,33,25,32,41,52,65,83,68,55,44,
%U A375510 35,34,43,54,67,82,102,85,70,57,46,36,45,56,69,84,101,123,104,87,72,59,48,47,58,71,86,103,122,146,125,106,89,74,61,49,60,73
%N A375510 Fringe indices of Zernike polynomials.
%C A375510 The Fringe indices reference the double indexed Zernike polynomials with a single ordinal. Although the set of Fringe indices is limited in practical applications, the mapping covers the entire set of polynomials.
%D A375510 Jim Schwiegerling, "Optical Specification, Fabrication, and Testing", SPIE, 2014, p. 90.
%H A375510 Gerhard Ramsebner, <a href="/A375510/b375510.txt">Table of n, a(n) for n = 0..10000</a>
%H A375510 Gerhard Ramsebner, <a href="/A375510/a375510.svg">animated SVG</a>
%H A375510 Gerhard Ramsebner, <a href="/A375510/a375510.pdf">PDF</a>
%H A375510 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zernike_polynomials#Fringe/University_of_Arizona_indices">Fringe / University of Arizona indices</a>
%H A375510 <a href="/index/Per#IntegerPermutation">Index to sequences related to the permutation of the positive integers</a>
%F A375510 T(n,k) = (1 + (n + abs(m))/2)^2 - 2*abs(m) + [m < 0], where m = -n+2*k and [] is the Iverson bracket.
%e A375510                     (0,0)                            1
%e A375510                (1,-1)  (1,1)                       3   2
%e A375510           (2,-2)   (2,0)   (2,2)                 6   4   5
%e A375510      (3,-3)    (3,-1)  (3,1)   (3,3)          11   8   7   10
%e A375510 (4,-4)   (4,-2)    (4,0)   (4,2)   (4,4)   18   13   9   12   17
%o A375510 (PARI) T(n,k)=my(m=-n+2*k); (1 + (n + abs(m))/2)^2 - 2*abs(m) + (m < 0) \\ _Andrew Howroyd_, Aug 27 2024
%Y A375510 Cf. A176988.
%K A375510 nonn,easy,tabl
%O A375510 0,2
%A A375510 _Gerhard Ramsebner_, Aug 25 2024