%I #16 Oct 17 2024 08:22:05

%S 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 21,27,31,32,30,33,29,34,28,35,40,41,39,42,38,43,37,44,36,50,49,51,48,

%U 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 Noll index series of Zernike polynomials converted to ANSI index.

%C ANSI indices of Zernike polynomials sorted by Noll index.

%H Gerhard Ramsebner, <a href="/A375779/b375779.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Gerhard Ramsebner, <a href="/A375779/a375779.svg">Noll index of Zernike polynomials (animated SVG)</a>

%H Gerhard Ramsebner, <a href="/A375779/a375779.pdf">PDF</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Zernike_polynomials#Noll&#39;s_sequential_indices">Noll's sequential indices</a>

%F a(j) = (n(n+2)+m)/2 where n=floor((sqrt(8*(j-1)+1)-1)/2) and m=(-1)^j*(mod(n,2)+2*floor((j-n*(n+1)/2-1+mod(n+1,2))/2))

%e Noll indices ANSI indices

%e 1 0

%e 3 2 1 2

%e 5 4 6 3 4 5

%e 9 7 8 10 6 7 8 9

%e 15 13 11 12 14 10 11 12 13 14

%e ... ...

%o (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 Cf. A176988.

%K nonn,tabl,easy

%O 1,2

%A _Gerhard Ramsebner_, Aug 27 2024