OFFSET
0,2
COMMENTS
If X is an n-set and Y a fixed (n-4)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Numbers m >= 0 such that 8m+49 is a square. - Bruce J. Nicholson, Jul 28 2017
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions.
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 3.
Leo Tavares, Diamond illustration.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(4-3*x)/(1-x)^3.
a(n) = A126890(n,3) for n>2. - Reinhard Zumkeller, Dec 30 2006
a(n) = A028563(n)/2. - Zerinvary Lajos, Feb 12 2007
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,4), for n>=1. - Milan Janjic, Dec 20 2008
a(n) = n + a(n-1) + 3 (with a(0)=0). - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+3). - Gary Detlefs, Aug 10 2010
Sum_{n>=1} 1/a(n) = 363/490. - R. J. Mathar, Jul 14 2012
a(n) = 4n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=4..n+3} i. - Wesley Ivan Hurt, Jun 28 2013
E.g.f.: (1/2)*x*(x+8)*exp(x). - G. C. Greubel, Jul 13 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/7 - 319/1470. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = 15*cos(sqrt(57)*Pi/2)/(8*Pi).
Product_{n>=1} (1 + 1/a(n)) = -63*cos(sqrt(41)*Pi/2)/(8*Pi). (End)
MATHEMATICA
Table[n*(n + 7)/2, {n, 0, 50}] (* G. C. Greubel, Jul 13 2017 *)
PROG
(PARI) a(n)=n*(n+7)/2 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jun 16 2000
EXTENSIONS
More terms from James A. Sellers, Jul 04 2000
STATUS
approved