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A167875
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One third of product plus sum of three consecutive nonnegative integers.
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7
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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
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OFFSET
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0,2
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COMMENTS
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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 [From Bruno Berselli, May 18 2010]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = (n^3+3*n^2+5*n+3)/3 for n > 0.
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). [From 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. [From Bruno Berselli, May 26 2010]
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EXAMPLE
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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 . [From Bruno Berselli, May 26 2010]
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MATHEMATICA
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Select[Table[(n*(n+1)*(n+2)+n+(n+1)+(n+2))/3, {n, 0, 5!}], IntegerQ[#]&](*From Vladimir Joseph Stephan Orlovsky, 04 Dec 2010*)
(Times@@#+Total[#])/3&/@Partition[Range[0, 65], 3, 1] (* From Harvey P. Dale, Mar 14 2011 *)
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PROG
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(MAGMA) [ (&*s + &+s)/3 where s is [n..n+2]: n in [0..42] ];
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CROSSREFS
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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.
Sequence in context: A099074 A014818 A006527 * A057304 A001752 A160860
Adjacent sequences: A167872 A167873 A167874 * A167876 A167877 A167878
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KEYWORD
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nonn
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AUTHOR
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Klaus Brockhaus, Nov 14 2009
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STATUS
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approved
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