A062748 Fourth column (r=3) of FS(3) staircase array A062745.

3, 9, 19, 34, 55, 83, 119, 164, 219, 285, 363, 454, 559, 679, 815, 968, 1139, 1329, 1539, 1770, 2023, 2299, 2599, 2924, 3275, 3653, 4059, 4494, 4959, 5455, 5983, 6544, 7139, 7769, 8435, 9138, 9879, 10659, 11479, 12340, 13243, 14189, 15179, 16214, 17295, 18423

Offset

0,1

Comments

In the Frey-Sellers reference this sequence is called {(n+2) over 3}_{2}, n >= 0.

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

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=6, a(n-6)=coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 26 2010

For n>=4, a(n-4) is the number of permutations of 1,2,...,n, such that n-3 is the only up-point, or, the same, a(n-4) is up-down coefficient {n,4} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014

Links

Table of n, a(n) for n=0..45.

Guillaume Aupy and Julien Herrmann. Periodicity in optimal hierarchical checkpointing schemes for adjoint computations. Optimization Methods and Software, Volume 32, 2017 - Issue 3. Preprint

D. D. Frey and J. A. Sellers, Generalizing Bailey's generalization of the Catalan numbers, The Fibonacci Quarterly, 39 (2001) 142-148.

Milan Janjic, Two Enumerative Functions

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = A062745(n+2, 3) = binomial(n+4, 3) - 1 = (n+1)*(n^2 + 8*n + 18)/3!.

G.f.: N(3;1, x)/(1-x)^4 with N(3;1, x) = 3 - 3*x + x^2, polynomial of the second row of A062746.

a(n-3) = ((n^3 - n)/6) - 1, n >= 3. - Zerinvary Lajos, May 05 2007

a(n) = A000292(n+2) - 1. - Zerinvary Lajos, May 05 2007

From Artur Jasinski, Mar 14 2008: (Start)

a(n) = sum of n successive triangular numbers A000217 starting from n=2.

a(n) = Sum[i(i+1)/2,{i=2..n}]. (End)

a(n) = -A050407(-1-n) for all n in Z. - Michael Somos, Jan 28 2018

a(n) = A000292(n+3) - A000124(n+3). - Torlach Rush, Aug 03 2018

Example

G.f. = 3 + 9*x + 19*x^2 + 34*x^3 + 55*x^4 + 83*x^5 + 119*x^6 + 164*x^7 + ...

Maple

[seq(binomial(n, 3)-1, n=4..41)]; # Zerinvary Lajos, Nov 25 2006
a:=n->sum ((j+1)*j/2, j=2..n): seq(a(n), n=2..39); # Zerinvary Lajos, Dec 17 2006
seq(((n^3-n)/6)-1, n=3..40); # Zerinvary Lajos, May 05 2007
seq(sum(sum(sum(1, k=0..l), l=0..m), m=1..n), n=1..38); # Zerinvary Lajos, Jan 26 2008

Mathematica

k = 0; a = {}; Do[f = n(n + 1)/2; k = k + f; AppendTo[a, k], {n, 2, 100}]; a (* Artur Jasinski, Mar 14 2008 *)
LinearRecurrence[{4, -6, 4, -1}, {3, 9, 19, 34}, 40] (* Harvey P. Dale, Jan 13 2019 *)

Prog

(PARI) {a(n) = binomial(n+4, 3) - 1}; /* Michael Somos, Jan 28 2018 */

Crossrefs

A column of triangle A014473.

Cf. A050407.

Sequence in context: A155274 A058058 A348308 * A325666 A147174 A147158

Adjacent sequences: A062745 A062746 A062747 * A062749 A062750 A062751

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 12 2001

Status

approved