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A055998 a(n) = n*(n+5)/2. 57
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = A126890(n,2) for n>1. - Reinhard Zumkeller, Dec 30 2006

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

REFERENCES

A. 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

Milan Janjic, Two Enumerative Functions

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) = C(n,2) - 2*n, n>=5. - Zerinvary Lajos, Nov 25 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) = n + a(n-1) + 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

a(n) = A014695(n+2)*A178242(n). - Paul Curtz, Jan 16 2011

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

MATHEMATICA

f[n_]:=n*(n+5)/2; f[Range[0, 100]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

PROG

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

CROSSREFS

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

Cf. A000096, A000217, A001477, A002522.

Row n=2 of A255961.

Sequence in context: A095115 A141214 A027379 * A066379 A024517 A257941

Adjacent sequences:  A055995 A055996 A055997 * A055999 A056000 A056001

KEYWORD

nonn,easy

AUTHOR

Barry E. Williams, Jun 14 2000

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.