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Expansion of (1 - x + x^4) / (1 - x)^3.
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%I #29 Mar 08 2022 11:42:11

%S 1,2,3,4,6,9,13,18,24,31,39,48,58,69,81,94,108,123,139,156,174,193,

%T 213,234,256,279,303,328,354,381,409,438,468,499,531,564,598,633,669,

%U 706,744,783,823,864,906,949,993,1038,1084,1131,1179

%N Expansion of (1 - x + x^4) / (1 - x)^3.

%C For n>2, maximal number of edges in critical strongly connected digraphs on n-1 vertices.

%C If Y is a 3-subset of an n-set X then, for n>=3, a(n) is the number of 2-subsets of X which do not have exactly one element in common with Y. Also, if Y is a 3-subset of an n-set X then, for n>=4, a(n-3) is the number of (n-2)-subsets of X which have no exactly two elements in common with Y. - _Milan Janjic_, Dec 28 2007

%H G. C. Greubel, <a href="/A016028/b016028.txt">Table of n, a(n) for n = 1..5000</a>

%H R. Aharoni and E. Berger, <a href="http://arXiv.org/abs/math.CO/9911113">The number of edges in critical strongly connected graphs</a>, arXiv:math/9911113 [math.CO], 1999.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F Also, from the third term on, triangular numbers + 3. - _Alexandre Wajnberg_, Dec 10 2005

%F a(n) = binomial(n,2) - 3*n + 9, n>=3. a(n-3) = n^2/2 - 7*n/2 + 9, n>=4. - _Milan Janjic_, Dec 28 2007

%t i=0;s=3;lst={1, 2};Do[s+=n+i;AppendTo[lst, s], {n, 0, 6!, 1}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 30 2008 *)

%t CoefficientList[Series[(1-x+x^4)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1},{1,2,3,4,6},60] (* _Harvey P. Dale_, Nov 30 2015 *)

%o (PARI) Vec((1-x+x^4)/(1-x)^3+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012

%Y Essentially triangular numbers (A000217) plus 3. Cf. A000124.

%K nonn,easy

%O 1,2

%A _Robert G. Wilson v_

%E Definition corrected by _Harvey P. Dale_, Nov 30 2015