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A016028 Expansion of (1 - x + x^4) / (1 - x)^3.

%I A016028 #29 Mar 08 2022 11:42:11
%S A016028 1,2,3,4,6,9,13,18,24,31,39,48,58,69,81,94,108,123,139,156,174,193,
%T A016028 213,234,256,279,303,328,354,381,409,438,468,499,531,564,598,633,669,
%U A016028 706,744,783,823,864,906,949,993,1038,1084,1131,1179
%N A016028 Expansion of (1 - x + x^4) / (1 - x)^3.
%C A016028 For n>2, maximal number of edges in critical strongly connected digraphs on n-1 vertices.
%C A016028 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 A016028 G. C. Greubel, <a href="/A016028/b016028.txt">Table of n, a(n) for n = 1..5000</a>
%H A016028 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 A016028 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A016028 Also, from the third term on, triangular numbers + 3. - _Alexandre Wajnberg_, Dec 10 2005
%F A016028 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 A016028 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 A016028 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 A016028 (PARI) Vec((1-x+x^4)/(1-x)^3+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y A016028 Essentially triangular numbers (A000217) plus 3. Cf. A000124.
%K A016028 nonn,easy
%O A016028 1,2
%A A016028 _Robert G. Wilson v_
%E A016028 Definition corrected by _Harvey P. Dale_, Nov 30 2015