%I #26 Sep 08 2022 08:45:48
%S 1,4,11,24,45,76,119,176,249,340,451,584,741,924,1135,1376,1649,1956,
%T 2299,2680,3101,3564,4071,4624,5225,5876,6579,7336,8149,9020,9951,
%U 10944,12001,13124,14315,15576,16909,18316,19799,21360,23001,24724,26531
%N One third of product plus sum of three consecutive nonnegative integers; a(n)=(n+1)(n^2+2n+3)/3.
%C a(n) = ((n*(n+1)*(n+2))+(n+(n+1)+(n+2)))/3, n >= 0.
%C Equals A006527 without initial term 0: a(n) = A006527(n+1).
%C Binomial transform of A167876.
%C Inverse binomial transform of A080930.
%C a(n) = A007290(n+2)+n+1.
%C a(n) = A014820(n)/(n+1) for n > 0.
%C a(n) = A116731(n+2)-1.
%C a(n) = A033547(n+1)-n.
%C a(n) = A054602(n)/3.
%C a(n) = A086514(n+3)-2.
%C a(n) = A002061(n+1)+a(n-1) for n > 0.
%C a(n) = A005894(n)-a(n-1) for n > 0.
%C First bisection is A057813.
%C Second differences are in A004277.
%C a(n) = A177342(n)*(-1)+a(n-1)*5 with n>0. For n=8, a(8)=-A177342(8)+a(7)*5=-631+176*5=249. - _Bruno Berselli_, May 18 2010
%H Vincenzo Librandi, <a href="/A167875/b167875.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) = (n^3+3*n^2+5*n+3)/3.
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3)+2 for n > 3; a(0)=1, a(1)=4, a(2)=11, a(3)=24.
%F G.f.: (1+x^2)/(1-x)^4.
%F a(n) = SUM(A109613(k)*A005408(n-k): 0<=k<=n). - _Reinhard Zumkeller_, Dec 05 2009
%F a(n)-4*a(n-1)+6*a(n-2)-4*a(n-3)+a(n-4)=0 for n>3. - _Bruno Berselli_, May 26 2010
%e a(0) = (0*1*2+0+1+2)/3 = (0+3)/3 = 1.
%e a(1) = (1*2*3+1+2+3)/3 = (6+6)/3 = 4.
%e a(6)-4*a(5)+6*a(4)-4*a(3)+a(2) = 119-4*76+6*45-4*24+11 = 0. - _Bruno Berselli_, May 26 2010
%t Select[Table[(n*(n+1)*(n+2)+n+(n+1)+(n+2))/3,{n,0,5!}],IntegerQ[#]&] (* _Vladimir Joseph Stephan Orlovsky_, Dec 04 2010 *)
%t (Times@@#+Total[#])/3&/@Partition[Range[0,65],3,1] (* _Harvey P. Dale_, Mar 14 2011 *)
%o (Magma) [ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];
%o (PARI) a(n)=(n+1)*(n^2+2*n+3)/3 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A001477 (nonnegative integers),
%Y A006527 ((n^3+2*n)/3),
%Y A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...),
%Y A080930,
%Y A007290 (2*C(n, 3)),
%Y A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2),
%Y A116731,
%Y A033547 (n*(n^2+5)/3),
%Y A054602 (Sum_{d|3} phi(d)*n^(3/d)),
%Y A086514 ((n^3-6*n^2+14*n-6)/3),
%Y A002061 (n^2-n+1),
%Y A005894 (centered tetrahedral numbers),
%Y A057813 ((2*n+1)*(4*n^2+4*n+3)/3),
%Y A004277 (1 and the positive even numbers),
%Y A028387 (n+(n+1)^2),
%Y A166941, A166942, A166943.
%K nonn,easy
%O 0,2
%A _Klaus Brockhaus_, Nov 14 2009