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A069132 Centered 19-gonal numbers.

%I A069132 #53 Feb 16 2025 08:32:45
%S A069132 1,20,58,115,191,286,400,533,685,856,1046,1255,1483,1730,1996,2281,
%T A069132 2585,2908,3250,3611,3991,4390,4808,5245,5701,6176,6670,7183,7715,
%U A069132 8266,8836,9425,10033,10660,11306,11971,12655,13358,14080,14821,15581,16360,17158
%N A069132 Centered 19-gonal numbers.
%C A069132 Binomial transform of [1, 19, 19, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 19, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 28 2011
%H A069132 Ivan Panchenko, <a href="/A069132/b069132.txt">Table of n, a(n) for n = 1..1000</a>
%H A069132 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>
%H A069132 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H A069132 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A069132 a(n) = (19*n^2 - 19*n + 2)/2.
%F A069132 a(n) = 19*n + a(n-1) - 19 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F A069132 G.f.: x*(1 + 17*x + x^2) / (1-x)^3. - _R. J. Mathar_, Feb 04 2011
%F A069132 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=20, a(2)=58. - _Harvey P. Dale_, Aug 21 2011
%F A069132 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A069132 Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(11/19)*Pi/2)/sqrt(209).
%F A069132 Sum_{n>=1} a(n)/n! = 21*e/2 - 1.
%F A069132 Sum_{n>=1} (-1)^n * a(n)/n! = 21/(2*e) - 1. (End)
%F A069132 E.g.f.: exp(x)*(1 + 19*x^2/2) - 1. - _Nikolaos Pantelidis_, Feb 06 2023
%e A069132 a(5)= 191 because (19*5^2 - 19*5 + 2)/2 = (475 - 95 + 2)/2 = 382/2 = 191.
%t A069132 FoldList[#1 + #2 &, 1, 19 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A069132 Table[(19n^2-19n+2)/2,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,20,58},50] (* _Harvey P. Dale_, Aug 21 2011 *)
%o A069132 (PARI) a(n)=19*binomial(n,2)+1 \\ _Charles R Greathouse IV_, Jul 29 2011
%Y A069132 Cf. centered polygonal numbers listed in A069190.
%K A069132 easy,nice,nonn
%O A069132 1,2
%A A069132 _Terrel Trotter, Jr._, Apr 07 2002