OFFSET
0,2
COMMENTS
If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Bisection of A165157. - Jaroslav Krizek, Sep 05 2009
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012
Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017
a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017
a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020
Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
Milan Janjic, Two Enumerative Functions.
Kival Ngaokrajang, Illustration from A000027 (contains errors).
Linhui Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv:2003.07901 [math.RT], 2020. See p. 8.
Leo Tavares, Illustration: Truncated Point Triangles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: x*(3-2*x)/(1-x)^3.
a(n) = A027379(n), n > 0.
a(n) = A126890(n,2) for n > 1. - Reinhard Zumkeller, Dec 30 2006
If we define f(n,i,m) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-m-j), then a(n) = -f(n,n-1,3), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = A167544(n+8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n + 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+2). - Gary Detlefs, Aug 10 2010
Sum_{n>=1} 1/a(n) = 137/150. - R. J. Mathar, Jul 14 2012
a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=3..n+2} i. - Wesley Ivan Hurt, Jun 28 2013
a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - Andrey Zabolotskiy, May 18 2016
E.g.f.: x*(6+x)*exp(x)/2. - G. C. Greubel, Apr 05 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)
MATHEMATICA
f[n_]:=n*(n+5)/2; f[Range[0, 50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
PROG
(PARI) a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019
(Sage) [n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019
CROSSREFS
a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
Row n=2 of A255961.
Cf. other rows, columns and diagonals of A000027 written as a table: A034856, A046691, A052905, A055999, A155212, A051936, A056000, A183897, A056115, A051938; A000124, A022856, A152950, A145018, A077169, A166136, A167487, A173036; A059993, A090288, A054000, A142463, A056220, A001105, A001844, A058331, A051890, A097080, A093328, A137882.
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Jun 14 2000
STATUS
approved