A375510 - OEIS

%I #44 Sep 14 2024 06:53:26

%S 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 28,37,51,40,31,24,23,30,39,50,66,53,42,33,25,32,41,52,65,83,68,55,44,

%U 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 Fringe indices of Zernike polynomials.

%C 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 Jim Schwiegerling, "Optical Specification, Fabrication, and Testing", SPIE, 2014, p. 90.

%H Gerhard Ramsebner, <a href="/A375510/b375510.txt">Table of n, a(n) for n = 0..10000</a>

%H Gerhard Ramsebner, <a href="/A375510/a375510.svg">animated SVG</a>

%H Gerhard Ramsebner, <a href="/A375510/a375510.pdf">PDF</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Zernike_polynomials#Fringe/University_of_Arizona_indices">Fringe / University of Arizona indices</a>

%F 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                     (0,0)                            1

%e                (1,-1)  (1,1)                       3   2

%e           (2,-2)   (2,0)   (2,2)                 6   4   5

%e      (3,-3)    (3,-1)  (3,1)   (3,3)          11   8   7   10

%e (4,-4)   (4,-2)    (4,0)   (4,2)   (4,4)   18   13   9   12   17

%o (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 Cf. A176988.

%K nonn,easy,tabl

%O 0,2

%A _Gerhard Ramsebner_, Aug 25 2024