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

1, 2, 3, 4, 6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81, 94, 108, 123, 139, 156, 174, 193, 213, 234, 256, 279, 303, 328, 354, 381, 409, 438, 468, 499, 531, 564, 598, 633, 669, 706, 744, 783, 823, 864, 906, 949, 993, 1038, 1084, 1131, 1179

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• easy

Comments

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

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• R. Aharoni and E. Berger, The number of edges in critical strongly connected graphs, arXiv:math/9911113 [math.CO], 1999.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Programs

Mathematica

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 *)
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 *)

PARI

Vec((1-x+x^4)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

Also, from the third term on, triangular numbers + 3. - Alexandre Wajnberg, Dec 10 2005

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

Extensions

Definition corrected by Harvey P. Dale, Nov 30 2015