A002348 - OEIS

%I M0549 N0198 #29 Jan 29 2022 01:08:37

%S 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 96,105,96,120,128,120,144,144,144,171,180,168,192,210,192,231,240,

%U 216,264,276,256,294,300,288,336,351,324,360,384,360,420,435,384,465

%N Degree of rational Poncelet porism of n-gon.

%D Kerawala, S. M.; Poncelet Porism in Two Circles. Bull. Calcutta Math. Soc. 39, 85-105, 1947.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A002348/b002348.txt">Table of n, a(n) for n = 3..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PonceletsPorism.html">Poncelet's Porism</a>

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

%t Poncelet[ n_Integer /; n >= 3 ] := Module[ {p, a, i}, {p, a}=Transpose[ FactorInteger[ n ] ];

%t 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 (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 (Haskell)

%o a002348 n = product (zipWith d ps es) * 4 ^ e0 `div` 8 where

%o    d p e = (p ^ 2 - 1) * p ^ e

%o    e0 = if even n then head $ a124010_row n else 0

%o    es = map ((* 2) . subtract 1) $

%o             if even n then tail $ a124010_row n else a124010_row n

%o    ps = if even n then tail $ a027748_row n else a027748_row n

%o -- _Reinhard Zumkeller_, Mar 18 2012

%Y Cf. A027748, A124010.

%K nonn,nice

%O 3,2

%A _N. J. A. Sloane_

%E Extended with Mathematica program by _Eric W. Weisstein_