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a(n) = 7*n*(n+1)/2 - 5.
1

%I #31 Feb 20 2023 03:09:35

%S 2,16,37,65,100,142,191,247,310,380,457,541,632,730,835,947,1066,1192,

%T 1325,1465,1612,1766,1927,2095,2270,2452,2641,2837,3040,3250,3467,

%U 3691,3922,4160,4405,4657,4916,5182,5455,5735,6022,6316,6617,6925,7240,7562

%N a(n) = 7*n*(n+1)/2 - 5.

%H Vincenzo Librandi, <a href="/A166154/b166154.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A166146(n)+1.

%F a(n) = a(n-1)+7*n = 3*a(n-1) -3*a(n-2) +a(n-3).

%F G.f.: x*(-2-10*x+5*x^2)/(x-1)^3.

%F E.g.f.: (7/2)*((x^2 + 2*x - 5)*exp(x) + 5). - _G. C. Greubel_, May 01 2016

%F Sum_{n>=1} 1/a(n) = 1/5 + (2*Pi/sqrt(329))*tan(sqrt(47/7)*Pi/2). - _Amiram Eldar_, Feb 20 2023

%t Table[7 n (n + 1)/2 - 5, {n, 100}] (* or *) CoefficientList[Series[(- 2 - 10 x + 5 x^2) / (x - 1)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 13 2013 *)

%t LinearRecurrence[{3,-3,1}, {2,16,37}, 50] (* _G. C. Greubel_, May 01 2016 *)

%o (Magma) [7*n*(n+1)/2-5: n in [1..50]]; // _Vincenzo Librandi_, Sep 13 2013

%o (PARI) a(n)=7*n*(n+1)/2-5 \\ _Charles R Greathouse IV_, May 02 2016

%Y Cf. A166146.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Oct 08 2009

%E Definition replaced by polynomial, A-number corrected, formulas added by _R. J. Mathar_, Oct 12 2009