A168240 a(n) = 13*n^2 + 7*n + 1.

21, 67, 139, 237, 361, 511, 687, 889, 1117, 1371, 1651, 1957, 2289, 2647, 3031, 3441, 3877, 4339, 4827, 5341, 5881, 6447, 7039, 7657, 8301, 8971, 9667, 10389, 11137, 11911, 12711, 13537, 14389, 15267, 16171, 17101, 18057, 19039, 20047, 21081, 22141, 23227

Offset: 1

Author

A.K. Devaraj, Nov 21 2009

Keywords

• nonn
• easy

Comments

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.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A165806, A165808, A165809, A168235.

Programs

Mathematica

LinearRecurrence[{3, -3, 1}, {21, 67, 139}, 50] (* G. C. Greubel, Apr 09 2016 *)
Table[13n^2+7n+1,{n,50}] (* Harvey P. Dale, Mar 22 2019 *)

PARI

a(n)=13*n^2+7*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

From R. J. Mathar, Nov 23 2009: (Start)

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

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

E.g.f.: (13*x^2 + 20*x + 1)*exp(x). - G. C. Greubel, Apr 09 2016

Extensions

Edited, definition simplified, sequence extended beyond a(8) by R. J. Mathar, Nov 23 2009