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A172082
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a(n) = n*(n+1)*(6*n-5)/2.
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4
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0, 1, 21, 78, 190, 375, 651, 1036, 1548, 2205, 3025, 4026, 5226, 6643, 8295, 10200, 12376, 14841, 17613, 20710, 24150, 27951, 32131, 36708, 41700, 47125, 53001, 59346, 66178, 73515, 81375, 89776, 98736, 108273, 118405, 129150, 140526
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OFFSET
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0,3
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COMMENTS
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Generated by formula: n*(n+1)*(2*d*n-2*d+3)/6 with d=9.
This sequence is related to A051682 by a(n) = n*A051682(n) - Sum_{i=0..n-1} A051682(i); in fact this is the case d=9 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n -2*d + 3)/6. - Bruno Berselli, Apr 16 2012
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REFERENCES
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E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - Bruno Berselli, Feb 13 2014
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LINKS
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Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
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FORMULA
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a(0)=0, a(1)=1, a(2)=21, a(3)=78; for n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jun 29 2011
a(n) = Sum_{i=0..n-1} (n-i)*(18*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(2 + 19*x + 6*x^2)*exp(x)/2. - G. C. Greubel, Aug 30 2019
Sum_{n>=1} 1/a(n) = 2*(3*sqrt(3)*Pi + 9*log(3) + 12*log(2) - 5)/55.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(6*Pi + 6*sqrt(3)*log(sqrt(3)+2) - 16*log(2) + 5)/55. (End)
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MAPLE
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MATHEMATICA
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Table[(18n^3+3n^2-15n)/6, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 21, 78}, 40] (* Harvey P. Dale, Jun 29 2011 *)
CoefficientList[Series[x*(1+17*x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 02 2014 *)
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PROG
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(PARI) vector(40, n, n*(n-1)*(6*n-11)/2) \\ G. C. Greubel, Aug 30 2019
(Sage) [n*(n+1)*(6*n-5)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> n*(n+1)*(6*n-5)/2); # G. C. Greubel, Aug 30 2019
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CROSSREFS
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Cf. similar sequences listed in A237616.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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