A162148 - OEIS

%I #36 Sep 08 2022 08:45:46

%S 0,4,17,44,90,160,259,392,564,780,1045,1364,1742,2184,2695,3280,3944,

%T 4692,5529,6460,7490,8624,9867,11224,12700,14300,16029,17892,19894,

%U 22040,24335,26784,29392,32164,35105,38220,41514,44992,48659,52520,56580

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

%C Partial sums of A147875.

%C Equals the fourth right hand column of A175136 for n>=1. - _Johannes W. Meijer_, May 06 2011

%C 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

%H Vincenzo Librandi, <a href="/A162148/b162148.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F From _Johannes W. Meijer_, May 06 2011: (Start)

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

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

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

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

%F 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

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

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

%t 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 *)

%o (Magma) [n*(n+1)*(5*n+7)/6: n in [0..50]]; // _Vincenzo Librandi_, May 07 2011

%o (PARI) a(n)=n*(n+1)*(5*n+7)/6 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Sage) [n*(n+1)*(5*n+7)/6 for n in (0..50)] # _G. C. Greubel_, Mar 31 2021

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

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

%Y Related to A000292 and A091894.

%K nonn,easy

%O 0,2

%A _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009

%E Definition rephrased by _R. J. Mathar_, Jun 27 2009