A055998 a(n) = n*(n+5)/2.

0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272

Offset: 0

Author

Barry E. Williams, Jun 14 2000

Keywords

• nonn
• easy

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).

Crossrefs

a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.

Cf. A000096, A000217, A001477, A002522.

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.

Programs

Magma

[n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019

Mathematica

f[n_]:=n*(n+5)/2; f[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

PARI

a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019

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

a(n) = A000217(n) + A005843(n). - Reinhard Zumkeller, Sep 24 2008

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

a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - Jaroslav Krizek, Sep 05 2009

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) = 3*A000217(n) - 2*A000217(n-1). - Bruno Berselli, Dec 17 2014

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)