A132769 - OEIS

%I #36 Nov 29 2024 21:13:27

%S 0,28,58,90,124,160,198,238,280,324,370,418,468,520,574,630,688,748,

%T 810,874,940,1008,1078,1150,1224,1300,1378,1458,1540,1624,1710,1798,

%U 1888,1980,2074,2170,2268,2368,2470,2574,2680,2788,2898,3010,3124,3240,3358,3478

%N a(n) = n*(n + 27).

%H Harvey P. Dale, <a href="/A132769/b132769.txt">Table of n, a(n) for n = 0..1000</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) + 26, with a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F a(0)=0, a(1)=28, a(2)=58; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Oct 14 2012

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

%F Sum_{n>=1} 1/a(n) = H(27)/27 = A001008(27)/A102928(27) = 312536252003/2168462696400, where H(k) is the k-th harmonic number.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/27 - 57128792093/2168462696400. (End)

%F From _Elmo R. Oliveira_, Nov 29 2024: (Start)

%F G.f.: 2*x*(14 - 13*x)/(1 - x)^3.

%F E.g.f.: exp(x)*x*(28 + x).

%F a(n) = 2*A132756(n). (End)

%t Table[n(n+27),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,28,58},50] (* _Harvey P. Dale_, Oct 14 2012 *)

%o (PARI) a(n)=n*(n+27) \\ _Charles R Greathouse IV_, Oct 07 2015

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

%Y Cf. A028569, A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759.

%Y Cf. A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768.

%Y Cf. A132756.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Aug 28 2007