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A375779 Noll index series of Zernike polynomials converted to ANSI index.

%I A375779 #22 Mar 27 2025 05:40:45
%S A375779 0,2,1,4,3,5,7,8,6,9,12,13,11,14,10,18,17,19,16,20,15,24,23,25,22,26,
%T A375779 21,27,31,32,30,33,29,34,28,35,40,41,39,42,38,43,37,44,36,50,49,51,48,
%U A375779 52,47,53,46,54,45,60,59,61,58,62,57,63,56,64,55,65,71,72,70,73,69,74,68,75,67,76,66,77
%N A375779 Noll index series of Zernike polynomials converted to ANSI index.
%C A375779 ANSI indices of Zernike polynomials sorted by Noll index.
%H A375779 Gerhard Ramsebner, <a href="/A375779/b375779.txt">Table of n, a(n) for n = 1..10000</a>
%H A375779 Robert J. Noll, <a href="https://doi.org/10.1364/JOSA.66.000207">Zernike polynomials and atmospheric turbulence</a>, J. Opt. Soc. Am. 1976, 66, 207-211.
%H A375779 Gerhard Ramsebner, <a href="/A375779/a375779.svg">Noll index of Zernike polynomials (animated SVG)</a>
%H A375779 Gerhard Ramsebner, <a href="/A375779/a375779.pdf">PDF</a>
%H A375779 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zernike_polynomials#Noll&#39;s_sequential_indices">Noll's sequential indices</a>
%F A375779 a(j) = (n(n+2)+m)/2 where n=floor( (sqrt(8*(j-1)+1)-1)/2 ) =A003056(j-1) and m = (-1)^j *( mod(n,2)+2*floor((j-n*(n+1)/2-1+mod(n+1,2))/2) ).
%F A375779 Quasi-inverse: A176988(1+a(n)) = n assuming offset 1 in A176988 and serialized format. - _R. J. Mathar_, Mar 26 2025
%e A375779 Noll indices     ANSI indices
%e A375779  1                0
%e A375779  3 2              1 2
%e A375779  5 4 6            3 4 5
%e A375779  9 7 8 10         6 7 8 9
%e A375779  15 13 11 12 14   10 11 12 13 14
%e A375779  ...              ...
%p A375779 A375779 := proc(j::integer)
%p A375779     n := floor((sqrt(8*(j-1)+1)-1)/2) ;
%p A375779     m := (-1)^j*(modp(n,2)+2*floor((j-n*(n+1)/2-1+modp(n+1,2))/2)) ;
%p A375779     (n*(n+2)+m)/2 ;
%p A375779 end proc:
%p A375779 seq(A375779(j),j=1..40) ; # _R. J. Mathar_, Mar 27 2025
%o A375779 (PARI) for(j=1, 28, my(n=floor((sqrt(8*(j-1)+1)-1)/2)); my(m=(-1)^j*(n%2+2*floor((j-n*(n+1)/2-1+(n+1)%2)/2))); print(j,",",(n*(n+2)+m)/2))
%Y A375779 Cf. A176988.
%K A375779 nonn,tabl,easy
%O A375779 1,2
%A A375779 _Gerhard Ramsebner_, Aug 27 2024