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A143208 a(1)=2; for n>1, a(n) = (4-9*n+3*n^2)/2.

%I A143208 #23 Jul 22 2024 15:17:03
%S A143208 2,-1,2,8,17,29,44,62,83,107,134,164,197,233,272,314,359,407,458,512,
%T A143208 569,629,692,758,827,899,974,1052,1133,1217,1304,1394,1487,1583,1682,
%U A143208 1784,1889,1997,2108,2222,2339,2459,2582,2708,2837,2969,3104,3242,3383,3527,3674
%N A143208 a(1)=2; for n>1, a(n) = (4-9*n+3*n^2)/2.
%C A143208 Old Name was: A sequence based on odd numbers of the type 3*n + 2: a(n) = a(n - 1) + n - 1; A000096; f(n) = 3*a(n)+2.
%H A143208 G. C. Greubel, <a href="/A143208/b143208.txt">Table of n, a(n) for n = 1..1000</a>
%H A143208 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A143208 From _Colin Barker_, Apr 14 2014: (Start)
%F A143208 a(n) = (4 - 9*n + 3*n^2)/2 for n>1.
%F A143208 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
%F A143208 G.f.: x*(2 - 7*x + 11*x^2 - 3*x^3)/ (1-x)^3. (End).
%F A143208 a(n) = (n-2)*A095794(n) - (n-1)*A095794(n-1) for n>1. [_Bruno Berselli_, May 19 2015]
%F A143208 E.g.f.: (1/2)*(4 - 6*x + 3*x^2)*exp(x) - 2 + 3*x. - _G. C. Greubel_, Jul 19 2024
%e A143208 G.f. = 2*x - x^2 + 2*x^3 + 8*x^4 + 17*x^5 + 29*x^6 + 44*x^7 + 62*x^8 + ...
%t A143208 a[0] = 0; a[1] = -1; a[n_] := a[n] = a[n - 1] + n - 1; a1 = Table[a[n], {n, 0, 30}]; f[n_] := 3*a[n] + 2; Table[f[n], {n, 0, 50}]
%t A143208 LinearRecurrence[{3,-3,1},{2,-1,2,8},60] (* _Harvey P. Dale_, Mar 22 2018 *)
%o A143208 (PARI) Vec(x*(3*x^3-11*x^2+7*x-2)/(x-1)^3 + O(x^100)) \\ _Colin Barker_, Apr 14 2014
%o A143208 (Magma) [n eq 1 select 2 else (4-9*n+3*n^2)/2: n in [1..60]]; // _G. C. Greubel_, Jul 19 2024
%o A143208 (SageMath) [(4-9*n+3*n^2)/2 + 3*int(n==1) for n in range(1,61)] # _G. C. Greubel_, Jul 19 2024
%Y A143208 Cf. A095794.
%K A143208 sign,easy
%O A143208 1,1
%A A143208 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 20 2008
%E A143208 Better name and edits by _Colin Barker_ and _Joerg Arndt_, Apr 14 2014