A060784 Number of double tangents of order n.

0, 4, 0, 0, 28, 120, 324, 700, 1320, 2268, 3640, 5544, 8100, 11440, 15708, 21060, 27664, 35700, 45360, 56848, 70380, 86184, 104500, 125580, 149688, 177100, 208104, 243000, 282100, 325728, 374220, 427924, 487200, 552420, 623968

Offset

0,2

References

H. Brocard and T. Lemoyne: Courbes géométriques remarquables (courbes spéciales) Planes et Gauches. Tome I, Paris: Albert Blanchard, 1967 [First publ. 1919]; see p. 375.

C.G.J. Jacobi, (Bericht ueber die zur Bekanntmachung geeigneten), Verhandlungen der Koenigl. Preuss. Akademie der Wiss. Berlin, 1850, p. 209, Jun 13, 1850. [Wolfdieter Lang, Oct 09 2001]

Links

Harry J. Smith, Table of n, a(n) for n = 0..1000

D. Ayala and R. Cavalieri, Counting bitangents with stable maps, arXiv:math/0505139 [math.AG], 2005.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*(n-2)*(n-3)*(n+3)/2.

From Colin Barker, Mar 16 2020: (Start)

G.f.: 4*x*(1 - 5*x + 10*x^2 - 3*x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 4, 0, 0, 28}, 40] (* Harvey P. Dale, Jul 25 2024 *)

Prog

(PARI) a(n)={n*(n - 2)*(n - 3)*(n + 3)/2} \\ Harry J. Smith, Jul 11 2009
(PARI) concat(0, Vec(4*x*(1 - 5*x + 10*x^2 - 3*x^3) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Mar 16 2020

Crossrefs

Sequence in context: A232357 A196302 A307186 * A181204 A191417 A307050

Adjacent sequences: A060781 A060782 A060783 * A060785 A060786 A060787

Keyword

nonn,easy

Author

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 28 2001

Status

approved