%I #39 Aug 01 2024 11:49:26

%S 1,23,67,133,221,331,463,617,793,991,1211,1453,1717,2003,2311,2641,

%T 2993,3367,3763,4181,4621,5083,5567,6073,6601,7151,7723,8317,8933,

%U 9571,10231,10913,11617,12343,13091,13861,14653,15467,16303,17161,18041,18943,19867,20813

%N Centered 22-gonal numbers.

%H Ivan Panchenko, <a href="/A069173/b069173.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.

%H R. Yin, J. Mu, and T. Komatsu, <a href="https://doi.org/10.20944/preprints202407.2280.v1">The p-Frobenius Number for the Triple of the Generalized Star Numbers</a>, Preprints 2024, 2024072280. See p. 2.

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.

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

%F a(n) = 11*n^2 - 11*n + 1.

%F a(n) = 22*n+a(n-1)-22 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010

%F From _Amiram Eldar_, Jun 21 2020: (Start)

%F Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7/11)*Pi/2)/sqrt(77).

%F Sum_{n>=1} a(n)/n! = 12*e - 1.

%F Sum_{n>=1} (-1)^n * a(n)/n! = 12/e - 1. (End)

%F E.g.f.: exp(x)*(1 + 11 * x^2) - 1. - _Nikolaos Pantelidis_, Feb 06 2023

%e a(5) = 221 because 11*5^2 - 11*5 + 1 = 275 - 55 + 1 = 221

%e For n=2, a(2)=22*2+1-22=23; n=3, a(3)=22*3+23-22=67; n=4, a(4)=22*4+67-22=133.

%t FoldList[#1 + #2 &, 1, 22 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)

%o (PARI) a(n)=11*n^2-11*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. centered polygonal numbers listed in A069190.

%K easy,nonn

%O 1,2

%A _Terrel Trotter, Jr._, Apr 09 2002