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%I #42 Feb 06 2023 14:00:42
%S 1,2,3,10,29,66,127,218,345,514,731,1002,1333,1730,2199,2746,3377,
%T 4098,4915,5834,6861,8002,9263,10650,12169,13826,15627,17578,19685,
%U 21954,24391,27002,29793,32770,35939,39306,42877,46658,50655,54874,59321
%N a(n) = (n-1)*(n-2)*(n-3) + n.
%C (n*a(n+1)^3+1)/(n^3+1) is the smallest integer of the form (n*k^3+1)/(n^3+1). - _Benoit Cloitre_, May 02 2002
%H Vincenzo Librandi, <a href="/A034324/b034324.txt">Table of n, a(n) for n = 1..1000</a>
%H Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.1007/s11139-021-00418-7">On k-partitions of multisets with equal sums</a>, The Ramanujan J. (2021) Vol. 55, 421-435.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = (n-2)^3 + 2 = A084380(n-2). - _Philippe Deléham_, Feb 23 2014
%F a(n+1) = A002061(n)*(n-2) + 3. - _Philippe Deléham_, Feb 23 2014
%F G.f.: x*(1-2*x+x^2+6*x^3)/(1-x)^4. - _Philippe Deléham_, Feb 23 2014
%F E.g.f.: 6 + (x^3-3*x^2+7*x-6)*exp(x). - _Nikolaos Pantelidis_, Feb 06 2023
%p seq( (n-2)^3 +2, n=1..50); # _G. C. Greubel_, Aug 23 2019
%t Table[(n-3)(n-2)(n-1)+n, {n,50}] (* or *) Table[n^3+2, {n,-1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 15 2011 *)
%t CoefficientList[Series[(1 -2x +x^2 +6x^3)/(1-x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Feb 24 2014 *)
%t LinearRecurrence[{4,-6,4,-1},{1,2,3,10},50] (* _Harvey P. Dale_, Aug 06 2018 *)
%o (PARI) a(n)=(n-3)*(n-2)*(n-1)+n \\ _Charles R Greathouse IV_, Jul 02 2016
%o (Magma) [(n-2)^3 +2: n in [1..50]]; // _G. C. Greubel_, Aug 23 2019
%o (Sage) [(n-2)^3 +2 for n in (1..50)] # _G. C. Greubel_, Aug 23 2019
%o (GAP) List([1..50], n-> (n-2)^3 +2); # _G. C. Greubel_, Aug 23 2019
%Y Cf. A000578, A084378, A084381.
%K easy,nonn
%O 1,2
%A Laurence Michaels (guardian(AT)ntplx.net)
%E Extended and corrected by _Erich Friedman_
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