|
%I #34 Sep 08 2022 08:44:39
%S 0,1,2,3,5,10,21,42,78,135,220,341,507,728,1015,1380,1836,2397,3078,
%T 3895,4865,6006,7337,8878,10650,12675,14976,17577,20503,23780,27435,
%U 31496,35992,40953,46410,52395,58941,66082,73853,82290,91430,101311
%N Number of intersection points of diagonals of an n-gon in general position, plus number of vertices.
%C If Y is a 3-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X which have neither one element nor two elements in common with Y; a(n-3) is then also the number of (n-4)-subsets of X which have neither one element nor two elements in common with Y. - _Milan Janjic_, Dec 28 2007
%H Vincenzo Librandi, <a href="/A014626/b014626.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (n^4 - 6*n^3 + 11*n^2 + 18*n)/24.
%F From _Paul Barry_, Sep 23 2004: (Start)
%F Binomial transform of (0, 1, 0, 0, 1, 0, 0, 0, ...), or g.f. x+x^4.
%F G.f.: x*(1-3*x+3*x^2)/(1-x)^5;
%F a(n) = C(n,1) + C(n,4). (End)
%F E.g.f.: x*(24 + x^3)*exp(x)/24. - _G. C. Greubel_, Nov 08 2018
%t Table[(n^4 -6*n^3 +11*n^2 +18*n)/24, {n, 0, 50}] (* _G. C. Greubel_, Nov 08 2018 *)
%o (Magma) [(n^4-6*n^3+11*n^2-6*n)/24 +n: n in [0..50]]; // _Vincenzo Librandi_, Aug 21 2011
%o (PARI) vector(50, n, n--; (n^4 -6*n^3 +11*n^2 +18*n)/24) \\ _G. C. Greubel_, Nov 08 2018
%K nonn,easy
%O 0,3
%A _Mohammad K. Azarian_
%E Corrected and extended by _Erich Friedman_
|
