A162148 a(n) = n*(n+1)*(5*n+7)/6.

0, 4, 17, 44, 90, 160, 259, 392, 564, 780, 1045, 1364, 1742, 2184, 2695, 3280, 3944, 4692, 5529, 6460, 7490, 8624, 9867, 11224, 12700, 14300, 16029, 17892, 19894, 22040, 24335, 26784, 29392, 32164, 35105, 38220, 41514, 44992, 48659, 52520, 56580

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Keywords

• nonn
• easy

Comments

Partial sums of A147875.

Equals the fourth right hand column of A175136 for n>=1. - Johannes W. Meijer, May 06 2011

a(n) is the number of triples (w,x,y) havingt all terms in {0,...,n} and x+y>w. - Clark Kimberling, Jun 14 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002412, A002413, A006331, A016061, A033994.

Cf. A162147, A175136, A190048, A190049, A254407.

Related to A000292 and A091894.

Programs

Magma

[n*(n+1)*(5*n+7)/6: n in [0..50]]; // Vincenzo Librandi, May 07 2011

Maple

A162148:= n-> n*(n+1)*(5*n+7)/6; seq(A162148(n), n=0..50); # G. C. Greubel, Mar 31 2021

Mathematica

Table[(n(n+1)(5n+7))/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,4,17,44}, 50] (* Harvey P. Dale, May 20 2014 *)

PARI

a(n)=n*(n+1)*(5*n+7)/6 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[n*(n+1)*(5*n+7)/6 for n in (0..50)] # G. C. Greubel, Mar 31 2021

Formula

a(n) = A162147(n) + A000217(n).

From Johannes W. Meijer, May 06 2011: (Start)

G.f.: x*(4+x)/(1-x)^4.

a(n) = 4*binomial(n+2,3) + binomial(n+1,3).

a(n) = A091894(3,0)*binomial(n+2,3) + A091894(3,1)*binomial(n+1,3). (End)

a(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i).

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4), a(0)=0, a(1)=4, a(2)=17, a(3)=44. - Harvey P. Dale, May 20 2014

E.g.f.: x*(24 +27*x +5*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021

Extensions

Definition rephrased by R. J. Mathar, Jun 27 2009