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A069178 Centered 21-gonal numbers.

%I A069178 #52 Feb 16 2025 08:32:45
%S A069178 1,22,64,127,211,316,442,589,757,946,1156,1387,1639,1912,2206,2521,
%T A069178 2857,3214,3592,3991,4411,4852,5314,5797,6301,6826,7372,7939,8527,
%U A069178 9136,9766,10417,11089,11782,12496,13231,13987,14764,15562,16381,17221,18082,18964
%N A069178 Centered 21-gonal numbers.
%H A069178 Ivan Panchenko, <a href="/A069178/b069178.txt">Table of n, a(n) for n = 1..1000</a>
%H A069178 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>
%H A069178 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H A069178 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A069178 a(n) = (21n^2 - 21n + 2)/2
%F A069178 a(n) = 21*n + a(n-1) - 21 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F A069178 G.f. -x*(1+19*x+x^2) / (x-1)^3. - _R. J. Mathar_, Feb 04 2011
%F A069178 Binomial transform of [1, 21, 21, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 21, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 26 2011
%F A069178 a(n) = 1 + Sum_{i=1..n} 21*(i-1). - _Wesley Ivan Hurt_, May 25 2013
%F A069178 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A069178 Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(13/21)*Pi/2)/sqrt(273).
%F A069178 Sum_{n>=1} a(n)/n! = 23*e/2 - 1.
%F A069178 Sum_{n>=1} (-1)^n * a(n)/n! = 23/(2*e) - 1. (End)
%F A069178 E.g.f.: exp(x)*(1 + 21*x^2/2)-1. - _Nikolaos Pantelidis_, Feb 06 2023
%t A069178 FoldList[#1 + #2 &, 1, 21 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A069178 LinearRecurrence[{3,-3,1},{1,22,64},60] (* _Harvey P. Dale_, Jun 13 2022 *)
%o A069178 (PARI) a(n)=(21*n^2-21*n+2)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A069178 Cf. centered polygonal numbers listed in A069190.
%K A069178 easy,nonn
%O A069178 1,2
%A A069178 _Terrel Trotter, Jr._, Apr 09 2002