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.

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, 26, 28, 35, 33, 31, 29, 30, 32, 34, 36, 45, 43, 41, 39, 37, 38, 40, 42, 44, 55, 53, 51, 49, 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

Offset: 0

Author

R. J. Mathar, Dec 08 2010

Keywords

• nonn
• easy
• tabl

Comments

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:

(0,0)

(1,-1) (1,1)

(2,-2) (2,0) (2,2)

(3,-3) (3,-1) (3,1) (3,3)

(4,-4) (4,-2) (4,0) (4,2) (4,4)

(5,-5) (5,-3) (5,-1) (5,1) (5,3) (5,5)

(6,-6) (6,-4) (6,-2) (6,0) (6,2) (6,4) (6,6)

(7,-7) (7,-5) (7,-3) (7,-1) (7,1) (7,3) (7,5) (7,7)

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

1,

3, 2,

5, 4, 6,

9, 7, 8,10,

15,13,11,12,14,

21,19,17,16,18,20,

27,25,23,22,24,26,28,

35,33,31,29,30,32,34,36,

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|.

Links

• N. Chetty and D. J. Griffith, Zernike-basis expansion of the fractional and radial Hilbert phase masks, Current Applied Physics, 15 (2015) 739-747
• R. J. Noll, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am 66 (1976) 207.
• Gerhard Ramsebner, Nollindex of the Zernike polynomials (animated SVG)
• Thomas Risse, Least Square Approximation with Zernike Polynomials Using SAGE, (2011).
• Wikipedia, Zernike Polynomials
• Index to sequences related to the permutation of the positive integers

Programs

Maple

Noll := proc(n,m)
    n*(n+1)/2+abs(m) ;
    if m>=0 and modp(n,4) in {2,3} then
        %+1 ;
    elif m<=0 and modp(n,4) in {0,1} then
        %+1 ;
    else
        % ;
    end if;
end proc:
A176988 := proc(n,k)
    Noll(n,-n+2*k) ;
end proc:
seq(seq(A176988(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Mar 27 2025

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)) );
row(n) = vector(n+1, k, A176988(n, k-1)); \\ Gerhard Ramsebner, Nov 10 2024

Formula

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