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%I #130 Jun 26 2023 10:40:19
%S 1,1,2,4,7,11,16,22,29,37,46,56,67,79,92,106,121,137,154,172,191,211,
%T 232,254,277,301,326,352,379,407,436,466,497,529,562,596,631,667,704,
%U 742,781,821,862,904,947,991,1036,1082,1129,1177,1226,1276,1327,1379
%N a(n) = 1 + (n-2)*(n-1)/2.
%C The sequence is the sum of upward sloping terms in an infinite lower triangle with 1's in the leftmost column and the odd integers in all other columns. - _Gary W. Adamson_, Jan 29 2014
%C For n > 1, if Kruskal's algorithm is run on a weighted connected graph of n nodes, then a(n) is the maximum number of iterations required to reach a spanning tree. - _Eric M. Schmidt_, Jun 04 2016
%C It can be observed that A152947/A000079, whose reduced numerators are A213671, is identical to its inverse binomial transform (except for signs); this shows that it is an "autosequence" (more precisely, an autosequence of the second kind). - _Jean-François Alcover_ (this remark is due to _Paul Curtz_), Jun 20 2016
%H Shawn A. Broyles, <a href="/A152947/b152947.txt">Table of n, a(n) for n = 1..1000</a>
%H Christian Bean, Bjarki Gudmundsson, and Henning Ulfarsson, <a href="https://arxiv.org/abs/1705.04109">Automatic discovery of structural rules of permutation classes</a>, arXiv:1705.04109 [math.CO], 2017.
%H Murat Ersen Berberler, Onur Ugurlu, and Gozde Kizilates, <a href="https://dergipark.org.tr/tr/download/article-file/190823">On a Subroutine for Covering Zeros in Hungarian Algorithm</a>, 2012, see section 5.1 on page 92.
%H H. Cheballah, S. Giraudo, and R. Maurice, <a href="http://arxiv.org/abs/1306.6605">Combinatorial Hopf algebra structure on packed square matrices</a>, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
%H Michael Dairyko, Samantha Tyner, Lara Pudwell, and Casey Wynn, <a href="https://doi.org/10.37236/2099">Non-contiguous pattern avoidance in binary trees</a>, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/notredame.pdf">Pattern avoidance in trees</a> (slides from a talk, mentions many sequences), 2012.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 1 + A000217(n-2) = A000124(n-2), n > 1. - _R. J. Mathar_, Jan 03 2009
%F a(n) = a(n-1) + n - 2 for n>1, a(1) = 1. - _Vincenzo Librandi_, Nov 26 2010
%F G.f.: -x*(1-2*x+2*x^2)/(x-1)^3. - _R. J. Mathar_, Nov 28 2010
%F From _Ilya Gutkovskiy_, Jun 04 2016: (Start)
%F E.g.f.: (4 - 2*x + x^2)*exp(x)/2 - 2.
%F Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) + 1 = A226985 + 1. (End)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - _Wesley Ivan Hurt_, Jun 20 2016
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - A228918. - _Amiram Eldar_, Jan 08 2023
%p A152947:=n->1+(n-2)*(n-1)/2: seq(A152947(n), n=1..100); # _Wesley Ivan Hurt_, Jun 20 2016
%t Table[1 + (n^2 - 3n + 2)/2, {n, 50}] (* _Alonso del Arte_, Jan 30 2014 *)
%o (Sage) [1+binomial(n,2) for n in range(0, 54)] # _Zerinvary Lajos_, Mar 12 2009
%o (Magma) [1+(n-2)*(n-1)/2: n in [1..60]]; // _Klaus Brockhaus_, Nov 28 2010
%o (PARI) a(n)=1+(n-2)*(n-1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000124, A000217, A226985.
%K nonn,easy
%O 1,3
%A _Vladimir Joseph Stephan Orlovsky_, Dec 15 2008
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