A002348 Degree of rational Poncelet porism of n-gon. (Formerly M0549 N0198)

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

3,2

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

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)

Example

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

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 ] ]

Prog

(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 */
(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

Crossrefs

Cf. A027748, A124010.

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

Sequence in context: A097273 A006446 A261342 * A019469 A081491 A386917

Adjacent sequences: A002345 A002346 A002347 * A002349 A002350 A002351

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Extended with Mathematica program by Eric W. Weisstein

Status

approved