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A005782 Number of n-gons in cubic curve.
(Formerly M5144)
2
24, 54, 216, 648, 2376, 8100, 29232, 104544, 381672, 1397070, 5163480, 19170432, 71587080, 268423200, 1010595960, 3817704744, 14467313448, 54975424194, 209430985176, 799644248064, 3059511345864, 11728121930100, 45035998958016, 173215362539520, 667199954727936 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 3..1000

M. Picquet, Applications de la representation des courbes du troisieme degre, Journal de l'École Polytechnique, Paris, 35 (1884), pp. 31-100.

FORMULA

When n is a prime Picquet gives a simple formula for a(n) - see A182589.

His formula for composite n is more complicated:

"Pour calculer le nombre propre des sommets des polygones de n côtés, on formera tous les diviseurs a de n complémentaires des diviseurs du même nombre qui n'admettent leurs facteurs premiers qu'à la première puissance, et si le nombre de ces facteurs est pair, on ajoutera à phi(n) ou chi(n), suivant que n est impair ou pair, les quantités phi(a) ou chi(a) suivant que a est impair ou pair; on les retranchera si les nombre des facteurs est impair."

"To calculate the number of vertices of the polygons with n sides we will get all the divisors a of n that are complementary divisors (codivisors) of the same number having their prime factors at the first power only, and if the number of these factors is even, we will add to phi(n) or chi(n), depending on whether n is even or odd, the quantities phi(a) or chi(a) depending on whether a is odd or even; we will subtract them if the number of factors is odd." - Michel Marcus, Feb 03 2013

MATHEMATICA

chi[n_] := (8*(2^(n - 1) + 1)*(2^(n - 2) - 1));

phi[n_] := (8*(2^(n - 2) + 1)*(2^(n - 1) - 1));

either[n_, a_, dsqf_] := (If [Mod[a, 2] == 0, v = chi[a], v = phi[a]]; If [a == n, v, If[Mod[PrimeNu[dsqf], 2] == 0, v, -v]]);

picquet[n_] := (ksum = 0; Do[If[SquareFreeQ[d], ksum += either[n, n/d, d]], {d, Divisors[n]}]; ksum/n);

Table[picquet[n], {n, 3, 27}] (* Jean-François Alcover, Mar 28 2016, after Michel Marcus *)

PROG

(PARI)

chi(n) = {return (8*(2^(n-1)+1)*(2^(n-2)-1)); }

phi(n) = {return (8*(2^(n-2)+1)*(2^(n-1)-1)); }

either(n, a, dsqf) = {if ((a % 2) == 0, v = chi(a), v = phi(a)); if (a == n, return (v)); if ((omega(dsqf) % 2) == 0, return (v), return (-v)); }

picquet(n) = {ksum = 0; fordiv(n, d, if (issquarefree(d), ksum += either(n, n/d, d)); ); return (ksum/n); }

/* Michel Marcus, Feb 03 2013 */

CROSSREFS

Cf. A182589.

Sequence in context: A322609 A234238 A228876 * A003756 A135191 A216697

Adjacent sequences:  A005779 A005780 A005781 * A005783 A005784 A005785

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Entry revised by N. J. A. Sloane, Nov 23 2011

More terms from Michel Marcus, Feb 03 2013

STATUS

approved

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Last modified October 21 14:20 EDT 2019. Contains 328301 sequences. (Running on oeis4.)