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A062026
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a(n) = n*(n+1)*(n^2 - 3*n + 6)/4.
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4
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0, 2, 6, 18, 50, 120, 252, 476, 828, 1350, 2090, 3102, 4446, 6188, 8400, 11160, 14552, 18666, 23598, 29450, 36330, 44352, 53636, 64308, 76500, 90350, 106002, 123606, 143318, 165300, 189720, 216752, 246576, 279378, 315350, 354690, 397602, 444296
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OFFSET
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0,2
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COMMENTS
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a(n) = 1*2*3 + 2*3*4 + 3*4*5 +. . .+ (n-2)*(n-1)*n +(n-1)*n*1+ n*1*2, the sum of the cyclic product of terms taken three at a time, final term being n*1*2=2n.
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LINKS
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FORMULA
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a(0)=0, a(1)=2, a(2)=6, a(3)=18, a(4)=50, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Apr 22 2015
a(n) = 6*binomial(n+3, 4) - 12*binomial(n+2, 3) + 8*binomial(n+1, 2).
G.f.: 2*x*(1 - 2*x + 4*x^2)/(1-x)^5.
E.g.f.: (1/4)*x*(8 + 4*x + 4*x^2 + x^3)*exp(x). (End)
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EXAMPLE
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a(4) = 1*2*3 + 2*3*4 + 3*4*1 + 4*1*2 = 50.
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MATHEMATICA
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Table[n(n+1)(n^2-3n+6)/4, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 6, 18, 50}, 40] (* Harvey P. Dale, Apr 22 2015 *)
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PROG
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(PARI) for (n=0, 100, write("b062026.txt", n, " ", n*(n+1)*(n^2 -3*n +6)/4) ) \\ Harry J. Smith, Jul 29 2009
(SageMath) [n*(n+1)*(n^2-3*n+6)/4 for n in (0..40)] # G. C. Greubel, May 05 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jun 06 2001
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STATUS
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approved
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