A167875 One third of product plus sum of three consecutive nonnegative integers; a(n)=(n+1)(n^2+2n+3)/3.

1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724, 26531

Offset: 0

Author

Klaus Brockhaus, Nov 14 2009

Keywords

• nonn
• easy

Comments

a(n) = ((n*(n+1)*(n+2))+(n+(n+1)+(n+2)))/3, n >= 0.

Equals A006527 without initial term 0: a(n) = A006527(n+1).

Binomial transform of A167876.

Inverse binomial transform of A080930.

a(n) = A007290(n+2)+n+1.

a(n) = A014820(n)/(n+1) for n > 0.

a(n) = A116731(n+2)-1.

a(n) = A033547(n+1)-n.

a(n) = A054602(n)/3.

a(n) = A086514(n+3)-2.

a(n) = A002061(n+1)+a(n-1) for n > 0.

a(n) = A005894(n)-a(n-1) for n > 0.

First bisection is A057813.

Second differences are in A004277.

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

Examples

			a(0) = (0*1*2+0+1+2)/3 = (0+3)/3 = 1.
a(1) = (1*2*3+1+2+3)/3 = (6+6)/3 = 4.
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
		

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. A001477 (nonnegative integers),

A006527 ((n^3+2*n)/3),

A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...),

A080930,

A007290 (2*C(n, 3)),

A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2),

A116731,

A033547 (n*(n^2+5)/3),

A054602 (Sum_{d|3} phi(d)*n^(3/d)),

A086514 ((n^3-6*n^2+14*n-6)/3),

A002061 (n^2-n+1),

A005894 (centered tetrahedral numbers),

A057813 ((2*n+1)*(4*n^2+4*n+3)/3),

A004277 (1 and the positive even numbers),

A028387 (n+(n+1)^2),

A166941, A166942, A166943.

Programs

Magma

[ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];

Mathematica

Select[Table[(n*(n+1)*(n+2)+n+(n+1)+(n+2))/3,{n,0,5!}],IntegerQ[#]&] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
(Times@@#+Total[#])/3&/@Partition[Range[0,65],3,1]  (* Harvey P. Dale, Mar 14 2011 *)

PARI

a(n)=(n+1)*(n^2+2*n+3)/3 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (n^3+3*n^2+5*n+3)/3.

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.

G.f.: (1+x^2)/(1-x)^4.

a(n) = SUM(A109613(k)*A005408(n-k): 0<=k<=n). - Reinhard Zumkeller, Dec 05 2009

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