A168240 - OEIS

%I #15 Dec 31 2023 15:17:13

%S 21,67,139,237,361,511,687,889,1117,1371,1651,1957,2289,2647,3031,

%T 3441,3877,4339,4827,5341,5881,6447,7039,7657,8301,8971,9667,10389,

%U 11137,11911,12711,13537,14389,15267,16171,17101,18057,19039,20047,21081,22141,23227

%N a(n) = 13*n^2 + 7*n + 1.

%C Consider the quadratic cyclotomic polynomial f(x) = x^2+x+1 and the quotients defined by f(x + n*f(x))/f(x). a(n) is the quotient at x=3.

%H G. C. Greubel, <a href="/A168240/b168240.txt">Table of n, a(n) for n = 1..5000</a>

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

%F From _R. J. Mathar_, Nov 23 2009: (Start)

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

%F G.f.: x*(21+4*x+x^2)/(1-x)^3. (End)

%F E.g.f.: (13*x^2 + 20*x + 1)*exp(x). - _G. C. Greubel_, Apr 09 2016

%e f(x) = 13 when x = 3. Hence at n = 1, f(x + f(x))/f(x) = 21 = a(1).

%t LinearRecurrence[{3, -3, 1}, {21, 67, 139}, 50] (* _G. C. Greubel_, Apr 09 2016 *)

%t Table[13n^2+7n+1,{n,50}] (* _Harvey P. Dale_, Mar 22 2019 *)

%o (PARI) a(n)=13*n^2+7*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A165806, A165808, A165809, A168235.

%K nonn,easy

%O 1,1

%A _A.K. Devaraj_, Nov 21 2009

%E Edited, definition simplified, sequence extended beyond a(8) by _R. J. Mathar_, Nov 23 2009