This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A102305 #35 Feb 04 2024 03:21:49
%S A102305 6,11,18,27,38,51,66,83,102,123,146,171,198,227,258,291,326,363,402,
%T A102305 443,486,531,578,627,678,731,786,843,902,963,1026,1091,1158,1227,1298,
%U A102305 1371,1446,1523,1602,1683,1766,1851,1938,2027,2118,2211,2306,2403
%N A102305 a(n) = n^2 + 2*n + 3.
%C A102305 Essentially a duplicate of A059100.
%H A102305 G. C. Greubel, <a href="/A102305/b102305.txt">Table of n, a(n) for n = 1..5000</a>
%H A102305 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A102305 a(n) = (1/5) * A027578(n-1).
%F A102305 a(n) = 2*n + a(n-1) + 1 (with a(1)=6). - _Vincenzo Librandi_, Nov 16 2010
%F A102305 a(n) = A059100(n+1). - _Reinhard Zumkeller_, Mar 21 2008
%F A102305 a(n) = A010000(n+1) for n >= 1. - _Georg Fischer_, Nov 02 2018
%F A102305 From _Amiram Eldar_, Sep 14 2022: (Start)
%F A102305 Sum_{n>=1} 1/a(n) = Pi * coth(sqrt(2)*Pi)/(2*sqrt(2)) - 7/12.
%F A102305 Sum_{n>=1} (-1)^(n+1)/a(n) = cosech(sqrt(2)*Pi)*Pi/(2*sqrt(2)) + 1/12. (End)
%F A102305 From _G. C. Greubel_, Feb 03 2024: (Start)
%F A102305 G.f.: (3 - 3*x + 2*x^2)/(1-x)^3.
%F A102305 E.g.f.: (3 + 3*x + x^2)*exp(x). (End)
%p A102305 A102305:=n->n^2+2*n+3: seq(A102305(n), n=1..100); # _Wesley Ivan Hurt_, Jan 22 2017
%t A102305 Table[n^2+2n+3,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{6,11,18},50] (* _Harvey P. Dale_, Aug 05 2015 *)
%o A102305 (PARI) a(n)=n^2+2*n+3 \\ _Charles R Greathouse IV_, Oct 16 2015
%o A102305 (Magma) [(n+1)^2+2: n in [1..60]]; // _G. C. Greubel_, Feb 03 2024
%o A102305 (SageMath) [n^2+2*n+3 for n in range(1,61)] # _G. C. Greubel_, Feb 03 2024
%Y A102305 Cf. A010000, A027578, A059100, A129388.
%K A102305 nonn,easy
%O A102305 1,1
%A A102305 _Ralf Stephan_, Jan 03 2005