cp's OEIS Frontend

A255848 a(n) = 2*n^2 + 18.

%I A255848 #67 Jan 25 2025 15:01:13
%S A255848 18,20,26,36,50,68,90,116,146,180,218,260,306,356,410,468,530,596,666,
%T A255848 740,818,900,986,1076,1170,1268,1370,1476,1586,1700,1818,1940,2066,
%U A255848 2196,2330,2468,2610,2756,2906,3060,3218,3380,3546,3716,3890,4068,4250,4436
%N A255848 a(n) = 2*n^2 + 18.
%C A255848 For n>3, the sequence gives the 6th diagonal of triangle in A055096.
%C A255848 Also, this is the case k=9 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. It is noted that a(n)*n = (n + sqrt(3))^3 + (n - sqrt(3))^3.
%C A255848 Equivalently, numbers m such that 2*m-36 is a square.
%H A255848 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A255848 a(n) = 2*A189834(n).
%F A255848 From _Vincenzo Librandi_, Mar 08 2015: (Start)
%F A255848 G.f.: 2*(9 - 17*x + 10*x^2)/(1 - x)^3.
%F A255848 a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A255848 From _Amiram Eldar_, Mar 28 2023: (Start)
%F A255848 Sum_{n>=0} 1/a(n) = (1 + 3*Pi*coth(3*Pi))/36.
%F A255848 Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/36. (End)
%F A255848 E.g.f.: 2*exp(x)*(9 + x + x^2). - _Elmo R. Oliveira_, Jan 25 2025
%t A255848 f[n_] := 2 n^2 + 18; Array[f, 50, 0] (* _Robert G. Wilson v_, Mar 08 2015 *)
%t A255848 CoefficientList[Series[(18 - 34 x + 20 x^2) / (1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 08 2015 *)
%t A255848 LinearRecurrence[{3,-3,1},{18,20,26},50] (* _Harvey P. Dale_, Aug 20 2021 *)
%o A255848 (Magma) [2*n^2+18: n in [0..50]]; // _Vincenzo Librandi_, Mar 08 2015
%o A255848 (PARI) vector(50, n, 2*n^2+18) \\ _Derek Orr_, Mar 09 2015
%o A255848 (Sage) [2*n^2+18 for n in (0..50)] # _Bruno Berselli_, Mar 11 2015
%Y A255848 Cf. A016825 (first differences), A055096, A189834.
%Y A255848 Subsequence of A047463.
%Y A255848 Cf. similar sequences listed in A255843.
%K A255848 nonn,easy
%O A255848 0,1
%A A255848 _Avi Friedlich_, Mar 08 2015
%E A255848 Edited by _Bruno Berselli_, Mar 11 2015