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Fourth diagonal (m=3) of triangle A084938; a(n) = A084938(n+3,n) = (n^3 + 9*n^2 + 26*n)/6.
2

%I #37 Sep 02 2017 00:15:37

%S 0,6,16,31,52,80,116,161,216,282,360,451,556,676,812,965,1136,1326,

%T 1536,1767,2020,2296,2596,2921,3272,3650,4056,4491,4956,5452,5980,

%U 6541,7136,7766,8432,9135,9876,10656,11476,12337,13240,14186,15176

%N Fourth diagonal (m=3) of triangle A084938; a(n) = A084938(n+3,n) = (n^3 + 9*n^2 + 26*n)/6.

%C If X is an n-set and Y a fixed (n-4)-subset of X then a(n-4) is equal to the number of 3-subsets of X intersecting Y. - _Milan Janjic_, Aug 15 2007

%C For n>=0, A092286(n) is the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = n + 1. A092286(n) is also the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 3n - 1. - _Clark Kimberling_, Mar 20 2012

%H Guillaume Aupy, Julien Herrmann. <a href="http://dx.doi.org/10.1080%2F10556788.2016.1230612">Periodicity in optimal hierarchical checkpointing schemes for adjoint computations</a>. Optimization Methods and Software, Volume 32, 2017 - Issue 3. <a href="http://people.bordeaux.inria.fr/gaupy/ressources/pub/journals/oms_periodicity.pdf">Preprint</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%F a(n) = A084938(n+3, n) = sum_{k=0..3} A090238(3, k)*binomial(n, k).

%F From _Gary Detlefs_, Aug 02 2010: (Start)

%F a(n) = 1/2 * sum_{k=1..n} (k+3)(k+2).

%F a(n) = 1/6 * n *(n^2 + 9n + 26). (End)

%F G.f.: x*(6 - 8*x + 3*x^2)/(1-x)^4. - _Colin Barker_, Mar 18 2012

%p a:=n->(n^3 + 9*n^2 + 26*n)/6: seq(a(n), n=3..45);

%t q=60;(Transpose[NestList[Accumulate,Range[q],q]]-Range[q])[[4]] (* _Vladimir Joseph Stephan Orlovsky_, Apr 08 2011 *)

%t Table[(n^3 + 9*n^2 + 26*n)/6, {n, 0, 100}] (* _T. D. Noe_, Apr 12 2011 *)

%Y Cf. A084938, A090238.

%K easy,nonn

%O 0,2

%A _Philippe Deléham_, Jan 30 2004