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A152947
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a(n) = 1 + (n-2)*(n-1)/2.
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20
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1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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
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
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
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LINKS
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Shawn A. Broyles, Table of n, a(n) for n = 1..1000
Christian Bean, Bjarki Gudmundsson, Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
Michael Dairyko, Samantha Tyner, Lara Pudwell, Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
Lara Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 1 + A000217(n-2) = A000124(n-2), n > 1. - R. J. Mathar, Jan 03 2009
a(n) = a(n-1) + n - 2 for n>1, a(1) = 1. - Vincenzo Librandi, Nov 26 2010
G.f.: -x*(1-2*x+2*x^2)/(x-1)^3. - R. J. Mathar, Nov 28 2010
From Ilya Gutkovskiy, Jun 04 2016: (Start)
E.g.f.: (4 - 2*x + x^2)*exp(x)/2 - 2.
Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) + 1 = A226985 + 1. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3. - Wesley Ivan Hurt, Jun 20 2016
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MAPLE
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A152947:=n->1+(n-2)*(n-1)/2: seq(A152947(n), n=1..100); # Wesley Ivan Hurt, Jun 20 2016
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MATHEMATICA
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Table[1 + (n^2 - 3n + 2)/2, {n, 50}] (* Alonso del Arte, Jan 30 2014 *)
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PROG
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(Sage) [1+binomial(n, 2) for n in xrange(0, 54)] # Zerinvary Lajos, Mar 12 2009
(MAGMA) [1+(n-2)*(n-1)/2: n in [1..60]]; // Klaus Brockhaus, Nov 28 2010
(PARI) a(n)=1+(n-2)*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015
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CROSSREFS
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Cf. A000124, A000217, A226985.
Sequence in context: A025732 A025739 A000124 * A212369 A212368 A217838
Adjacent sequences: A152944 A152945 A152946 * A152948 A152949 A152950
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KEYWORD
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nonn,easy
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AUTHOR
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Vladimir Joseph Stephan Orlovsky, Dec 15 2008
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STATUS
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approved
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