A005701 Number of exterior points formed by extending diagonals of n-gon in general position. (Formerly M2968)

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, 139230, 155400, 172938, 191919, 212420, 234520, 258300, 283843

Offset

0,1

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.

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

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)

From Elmo R. Oliveira, Dec 03 2025: (Start)

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

E.g.f.: exp(x)*(x^4 + 18*x^3 + 90*x^2 + 132*x + 36)/12. (End)

Mathematica

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

Prog

(Magma) [(n+1)*(n+2)*(n+3)*(n+6)/12: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013
(PARI) a(n) = (n+1)*(n+2)*(n+3)*(n+6)/12; \\ Michel Marcus, Dec 16 2017

Crossrefs

A diagonal of the triangle in A179898.

Sequence in context: A034130 A117662 A050297 * A196236 A213482 A296267

Adjacent sequences: A005698 A005699 A005700 * A005702 A005703 A005704

Keyword

nonn,easy

Author

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

Status

approved