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A002348 Degree of rational Poncelet porism of n-gon.

%I A002348 M0549 N0198 #33 Jul 23 2025 01:03:17
%S A002348 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,
%T A002348 96,105,96,120,128,120,144,144,144,171,180,168,192,210,192,231,240,
%U A002348 216,264,276,256,294,300,288,336,351,324,360,384,360,420,435,384,465
%N A002348 Degree of rational Poncelet porism of n-gon.
%D A002348 Kerawala, S. M.; Poncelet Porism in Two Circles. Bull. Calcutta Math. Soc. 39, 85-105, 1947.
%D A002348 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002348 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002348 Reinhard Zumkeller, <a href="/A002348/b002348.txt">Table of n, a(n) for n = 3..10000</a>
%H A002348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PonceletsPorism.html">Poncelet's Porism</a>
%F A002348 From _Ridouane Oudra_, Jul 19 2025: (Start)
%F A002348 a(n) = (1/8) * Sum_{k=0..A007814(n)} A007434(n/2^k).
%F A002348 a(n) = (1/8) * Sum_{d|n} A209229(d)*A007434(n/d).
%F A002348 a(n) = (1/8) * Sum_{d|n} A328407(d)*mu(n/d).
%F A002348 a(n) = (n^2/8) * Prod_{p|n, p prime > 2} (1 - 1/p^2).
%F A002348 a(n) = (1/8) * 4^A007814(n) * A007434(A000265(n)).
%F A002348 a(n) = A007434(n)/A010724(n). (End)
%e A002348 For a triangle the degree is 1, thus a(3) = 1. - _Michael Somos_, Dec 07 2018
%t A002348 Poncelet[ n_Integer /; n >= 3 ] := Module[ {p, a, i}, {p, a}=Transpose[ FactorInteger[ n ] ];
%t A002348 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 ] ]
%o A002348 (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 */
%o A002348 (Haskell)
%o A002348 a002348 n = product (zipWith d ps es) * 4 ^ e0 `div` 8 where
%o A002348    d p e = (p ^ 2 - 1) * p ^ e
%o A002348    e0 = if even n then head $ a124010_row n else 0
%o A002348    es = map ((* 2) . subtract 1) $
%o A002348             if even n then tail $ a124010_row n else a124010_row n
%o A002348    ps = if even n then tail $ a027748_row n else a027748_row n
%o A002348 -- _Reinhard Zumkeller_, Mar 18 2012
%Y A002348 Cf. A027748, A124010.
%Y A002348 Cf. A007434, A209229, A007434, A328407, A007814, A000265, A008683, A010724.
%K A002348 nonn,nice
%O A002348 3,2
%A A002348 _N. J. A. Sloane_
%E A002348 Extended with Mathematica program by _Eric W. Weisstein_