cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Gerhard Ramsebner

Gerhard Ramsebner's wiki page.

Gerhard Ramsebner has authored 3 sequences.

A377850 Noll index series of Zernike polynomials converted to Fringe index.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 11, 10, 9, 12, 13, 17, 18, 14, 15, 19, 20, 26, 27, 16, 22, 21, 29, 28, 38, 37, 24, 23, 31, 30, 40, 39, 51, 50, 25, 32, 33, 41, 42, 52, 53, 65, 66, 34, 35, 43, 44, 54, 55, 67, 68, 82, 83, 36, 46, 45, 57, 56, 70, 69, 85, 84, 102, 101, 48, 47, 59, 58, 72, 71, 87, 86
Offset: 1

Views

Author

Gerhard Ramsebner, Nov 09 2024

Keywords

Comments

Fringe indices of Zernike polynomials sorted by Noll index.

Examples

			  Noll indices      Fringe indices
   1                 1
   3  2              3  2
   5  4  6           6  4 5
   9  7  8 10       11  8 7 10
  15 13 11 12 14    18 13 9 12 17
  ...               ...

Links

Crossrefs

Cf. A176988, A375510.

Programs

  • PARI
    A377850(j) = my(n=floor( (sqrt(8*(j-1)+1)-1)/2 ), m=(-1)^j*(n%2+2*floor((j-n*(n+1)/2-1+(n+1)%2)/2))); (1+(n+abs(m))/2)^2 -2*abs(m)+(m<0);

Formula

a(j) = (1+(n+abs(m))/2)^2-2*abs(m)+[m<0] where n=floor((sqrt(8*(j-1)+1)-1)/2), m=(-1)^j*(mod(n,2)+2*floor((j-n*(n+1)/2-1+mod(n+1,2))/2)) and [] is the Iverson bracket.
a(A176988(j)) = A375510(j) assuming offset 1 in all 3 sequences and serialized versions.

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

Original entry on oeis.org

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, 21, 27, 31, 32, 30, 33, 29, 34, 28, 35, 40, 41, 39, 42, 38, 43, 37, 44, 36, 50, 49, 51, 48, 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
Offset: 1

Views

Author

Gerhard Ramsebner, Aug 27 2024

Keywords

Comments

ANSI indices of Zernike polynomials sorted by Noll index.

Examples

			Noll indices     ANSI indices
 1                0
 3 2              1 2
 5 4 6            3 4 5
 9 7 8 10         6 7 8 9
 15 13 11 12 14   10 11 12 13 14
 ...              ...

Links

Crossrefs

Cf. A176988.

Programs

  • Maple
    A375779 := proc(j::integer)
    n := floor((sqrt(8*(j-1)+1)-1)/2) ;
    m := (-1)^j*(modp(n,2)+2*floor((j-n*(n+1)/2-1+modp(n+1,2))/2)) ;
    (n*(n+2)+m)/2 ;
end proc:
seq(A375779(j),j=1..40) ; # R. J. Mathar, Mar 27 2025
  • 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))

Formula

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) ).
Quasi-inverse: A176988(1+a(n)) = n assuming offset 1 in A176988 and serialized format. - R. J. Mathar, Mar 26 2025

A375510 Fringe indices of Zernike polynomials.

Original entry on oeis.org

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, 28, 37, 51, 40, 31, 24, 23, 30, 39, 50, 66, 53, 42, 33, 25, 32, 41, 52, 65, 83, 68, 55, 44, 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
Offset: 0

Views

Author

Gerhard Ramsebner, Aug 25 2024

Keywords

Comments

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.

Examples

			                    (0,0)                            1
               (1,-1)  (1,1)                       3   2
          (2,-2)   (2,0)   (2,2)                 6   4   5
     (3,-3)    (3,-1)  (3,1)   (3,3)          11   8   7   10
(4,-4)   (4,-2)    (4,0)   (4,2)   (4,4)   18   13   9   12   17

References

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

Links

Crossrefs

Cf. A176988.

Programs

  • 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

Formula

T(n,k) = (1 + (n + abs(m))/2)^2 - 2*abs(m) + [m < 0], where m = -n+2*k and [] is the Iverson bracket.

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