A241848 - OEIS

%I #23 Jan 18 2025 13:31:28

%S 18,19,22,27,34,43,54,67,82,99,118,139,162,187,214,243,274,307,342,

%T 379,418,459,502,547,594,643,694,747,802,859,918,979,1042,1107,1174,

%U 1243,1314,1387,1462,1539,1618,1699,1782,1867,1954,2043,2134,2227,2322,2419,2518

%N a(n) = n^2 + 18.

%H Vincenzo Librandi, <a href="/A241848/b241848.txt">Table of n, a(n) for n = 0..1000</a>

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

%F G.f.: (18 - 35*x + 19*x^2)/(1 - x)^3.

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

%F From _Amiram Eldar_, Nov 03 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + sqrt(18)*Pi*coth(sqrt(18)*Pi))/36.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(18)*Pi*cosech(sqrt(18)*Pi))/36. (End)

%F E.g.f.: exp(x)*(18 + x + x^2). - _Elmo R. Oliveira_, Nov 29 2024

%t Table[n^2 + 18, {n, 0, 60}]

%t LinearRecurrence[{3,-3,1},{18,19,22},60] (* _Harvey P. Dale_, Jan 18 2025 *)

%o (Magma) [n^2+18: n in [0..60]];

%o (PARI) a(n)=n^2+18 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. similar sequences listed in A114962.

%K nonn,easy,changed

%O 0,1

%A _Vincenzo Librandi_, May 01 2014