A002348 Degree of rational Poncelet porism of n-gon.

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 21, 24, 24, 32, 36, 36, 45, 48, 48, 60, 66, 64, 75, 84, 81, 96, 105, 96, 120, 128, 120, 144, 144, 144, 171, 180, 168, 192, 210, 192, 231, 240, 216, 264, 276, 256, 294, 300, 288, 336, 351, 324, 360, 384, 360, 420, 435, 384, 465

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Examples

			For a triangle the degree is 1, thus a(3) = 1. - _Michael Somos_, Dec 07 2018
		

References

• Kerawala, S. M.; Poncelet Porism in Two Circles. Bull. Calcutta Math. Soc. 39, 85-105, 1947.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 3..10000
• Eric Weisstein's World of Mathematics, Poncelet's Porism

Crossrefs

Cf. A027748, A124010.

Cf. A007434, A209229, A007434, A328407, A007814, A000265, A008683, A010724.

Programs

Haskell

a002348 n = product (zipWith d ps es) * 4 ^ e0 `div` 8 where
   d p e = (p ^ 2 - 1) * p ^ e
   e0 = if even n then head $ a124010_row n else 0
   es = map ((* 2) . subtract 1) $
            if even n then tail $ a124010_row n else a124010_row n
   ps = if even n then tail $ a027748_row n else a027748_row n
-- Reinhard Zumkeller, Mar 18 2012

Mathematica

Poncelet[ n_Integer /; n >= 3 ] := Module[ {p, a, i}, {p, a}=Transpose[ FactorInteger[ n ] ];
If[ p[[1]]==2, 4^a[[1]] Product[ p[[i]]^(2(a[[i]] - 1))(p[[i]]^2 - 1), {i, 2, Length[ p ]} ]/8, (* Else *) Product[ p[[i]]^(2(a[[i]] - 1))(p[[i]]^2 - 1), {i, Length[ p ]} ]/8 ] ]

PARI

{a(n) = my(p, e); if( n<3, 0, p=factor(n)~; e=p[2,]; p=p[1,]; if( p[1]==2, 4^e[1], 1) * prod(i=1 + (p[1]==2), length(p), p[i]^(2*(e[i] - 1)) * (p[i]^2 - 1)) / 8)}; /* Michael Somos, Dec 09 1999 */

Formula

From Ridouane Oudra, Jul 19 2025: (Start)

a(n) = (1/8) * Sum_{k=0..A007814(n)} A007434(n/2^k).

a(n) = (1/8) * Sum_{d|n} A209229(d)*A007434(n/d).

a(n) = (1/8) * Sum_{d|n} A328407(d)*mu(n/d).

a(n) = (n^2/8) * Prod_{p|n, p prime > 2} (1 - 1/p^2).

a(n) = (1/8) * 4^A007814(n) * A007434(A000265(n)).

a(n) = A007434(n)/A010724(n). (End)

Extensions

Extended with Mathematica program by Eric W. Weisstein