A005701 Number of exterior points formed by extending diagonals of n-gon in general position.

3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• easy

Comments

See Gouyou-Beauchamps for an interpretation in terms of closed paths in the first quadrant of the square grid.

References

• Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986.
• Dominique Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)
• Index entries for sequences related to Young tableaux.

Crossrefs

A diagonal of the triangle in A179898.

Programs

Magma

[(n+1)*(n+2)*(n+3)*(n+6)/12: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013

Mathematica

CoefficientList[Series[(x - 3) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

PARI

a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12; \\ Michel Marcus, Dec 16 2017

Formula

a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12.

G.f.: (x-3)/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009

From Amiram Eldar, May 17 2025: (Start)

Sum_{n>=0} 1/a(n) = 137/300.

Sum_{n>=0} (-1)^n/a(n) = 32*log(2)/5 - 1247/300. (End)