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a(n) = n*(n + 28).
4

%I #33 Mar 13 2022 05:50:40

%S 0,29,60,93,128,165,204,245,288,333,380,429,480,533,588,645,704,765,

%T 828,893,960,1029,1100,1173,1248,1325,1404,1485,1568,1653,1740,1829,

%U 1920,2013,2108,2205,2304,2405,2508,2613,2720,2829,2940,3053,3168,3285,3404,3525

%N a(n) = n*(n + 28).

%H G. C. Greubel, <a href="/A132770/b132770.txt">Table of n, a(n) for n = 0..5000</a>

%H Felix P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, Preprint on ResearchGate, March 2014.

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

%F a(n) = 2*n + a(n-1) + 27, with a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F From _Amiram Eldar_, Jan 16 2021: (Start)

%F Sum_{n>=1} 1/a(n) = H(28)/28 = A001008(28)/A102928(28) = 315404588903/2248776129600, where H(k) is the k-th harmonic number.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 7751493599/321253732800. (End)

%F G.f.: x*(29 - 27*x)/(1-x)^3. - _Harvey P. Dale_, Aug 03 2021

%F E.g.f.: x*(29 + x)*exp(x). - _G. C. Greubel_, Mar 13 2022

%t #(#+28)&/@Range[0,50] (* or *) LinearRecurrence[{3,-3,1},{0,29,60},50] (* _Harvey P. Dale_, Apr 30 2018 *)

%o (PARI) a(n)=n*(n+28) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Sage) [n*(n+28) for n in (0..50)] # _G. C. Greubel_, Mar 13 2022

%Y Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.

%Y Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761.

%Y Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Aug 28 2007