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 A167875 One third of product plus sum of three consecutive nonnegative integers; a(n)=(n+1)(n^2+2n+3)/3. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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). 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 EXAMPLE 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 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 *) PROG (MAGMA) [ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ]; (PARI) a(n)=(n+1)*(n^2+2*n+3)/3 \\ Charles R Greathouse IV, Oct 07 2015 CROSSREFS Cf. A001477 (nonnegative integers), A006527 ((n^3+2*n)/3), A167876 (1, 3, 4, 2, 0, 0, 0, 0, ...), A007290 (2*C(n, 3)), A014820 ((1/3)*(n^2+2*n+3)*(n+1)^2), 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), Sequence in context: A014818 A328684 A006527 * A057304 A001752 A160860 Adjacent sequences:  A167872 A167873 A167874 * A167876 A167877 A167878 KEYWORD nonn,easy AUTHOR Klaus Brockhaus, Nov 14 2009 STATUS approved

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Last modified December 7 09:33 EST 2019. Contains 329843 sequences. (Running on oeis4.)