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%I #149 Feb 12 2024 02:28:32
%S 0,3,7,12,18,25,33,42,52,63,75,88,102,117,133,150,168,187,207,228,250,
%T 273,297,322,348,375,403,432,462,493,525,558,592,627,663,700,738,777,
%U 817,858,900,943,987,1032,1078,1125,1173,1222,1272
%N a(n) = n*(n+5)/2.
%C 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
%C Bisection of A165157. - _Jaroslav Krizek_, Sep 05 2009
%C 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
%C Numbers m >= 0 such that 8m+25 is a square. - _Bruce J. Nicholson_, Jul 26 2017
%C 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
%C This is the triangle equivalent of A294249 which requires matchstick-built patterns for squares 1 X 1, 2 X 2, ..., n X n. - _John King_, Apr 04 2019
%C a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - _Stefano Spezia_, Mar 24 2020
%C 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
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
%H Ivan Panchenko, <a href="/A055998/b055998.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Kival Ngaokrajang, <a href="/A000027/a000027_2.pdf">Illustration</a> from A000027 (contains errors).
%H Linhui Shen, <a href="https://arxiv.org/abs/2003.07901">Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems</a>, arXiv:2003.07901 [math.RT], 2020. See p. 8.
%H Leo Tavares, <a href="/A055998/a055998.jpg">Illustration: Truncated Point Triangles</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(3-2*x)/(1-x)^3.
%F a(n) = A027379(n), n > 0.
%F a(n) = A126890(n,2) for n > 1. - _Reinhard Zumkeller_, Dec 30 2006
%F a(n) = A000217(n) + A005843(n). - _Reinhard Zumkeller_, Sep 24 2008
%F 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
%F a(n) = A167544(n+8). - _Philippe Deléham_, Nov 25 2009
%F a(n) = a(n-1) + n + 2 with a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F a(n) = Sum_{k=1..n} (k+2). - _Gary Detlefs_, Aug 10 2010
%F a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - _Jaroslav Krizek_, Sep 05 2009
%F Sum_{n>=1} 1/a(n) = 137/150. - _R. J. Mathar_, Jul 14 2012
%F a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F a(n) = Sum_{i=3..n+2} i. - _Wesley Ivan Hurt_, Jun 28 2013
%F a(n) = 3*A000217(n) - 2*A000217(n-1). - _Bruno Berselli_, Dec 17 2014
%F a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - _Andrey Zabolotskiy_, May 18 2016
%F E.g.f.: x*(6+x)*exp(x)/2. - _G. C. Greubel_, Apr 05 2019
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - _Amiram Eldar_, Jan 10 2021
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)
%t f[n_]:=n*(n+5)/2; f[Range[0,50]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 10 2011 *)
%o (PARI) a(n)=n*(n+5)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [n*(n+5)/2: n in [0..50]]; // _G. C. Greubel_, Apr 05 2019
%o (Sage) [n*(n+5)/2 for n in (0..50)] # _G. C. Greubel_, Apr 05 2019
%Y a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
%Y Cf. A000096, A000217, A001477, A002522.
%Y Row n=2 of A255961.
%Y 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.
%K nonn,easy
%O 0,2
%A _Barry E. Williams_, Jun 14 2000
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