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%I #60 May 31 2022 15:40:53
%S 1,18,52,103,171,256,358,477,613,766,936,1123,1327,1548,1786,2041,
%T 2313,2602,2908,3231,3571,3928,4302,4693,5101,5526,5968,6427,6903,
%U 7396,7906,8433,8977,9538,10116,10711,11323,11952,12598,13261,13941,14638,15352
%N Centered 17-gonal numbers: (17*n^2 - 17*n + 2)/2.
%C Equals binomial transform of [1, 17, 17, 0, 0, 0, ...]. - _Gary W. Adamson_, Mar 26 2010
%H Ivan Panchenko, <a href="/A069130/b069130.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 <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) = (17*n^2 - 17*n + 2)/2.
%F a(n) = 17*n + a(n-1) - 17 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F G.f.: x*(1+15*x+x^2) / (1-x)^3. - _R. J. Mathar_, Feb 04 2011
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=52. - _Harvey P. Dale_, Jun 05 2011
%F Narayana transform (A001263) of [1, 17, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 28 2011
%F From _Amiram Eldar_, Jun 21 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 2*Pi*tan(3*Pi/(2*sqrt(17)))/(3*sqrt(17)).
%F Sum_{n>=1} a(n)/n! = 19*e/2 - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 19/(2*e) - 1. (End)
%F E.g.f.: exp(x)*(1 + 17*x^2/2) - 1. - _Stefano Spezia_, May 31 2022
%e a(5) = 171 because (17*5^2 - 17*5 + 2)/2 = (425 - 85 + 2)/2 = 342/2 = 171.
%p A069130:=n->(17*n^2 - 17*n + 2)/2; seq(A069130(n), n=1..50); # _Wesley Ivan Hurt_, Jun 09 2014
%t FoldList[#1 + #2 &, 1, 17 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t Table[(17n^2-17n+2)/2,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,18,52},50] (* _Harvey P. Dale_, Jun 05 2011 *)
%o (PARI) a(n)=17*binomial(n,2)+1 \\ _Charles R Greathouse IV_, Jun 05 2011
%o (Magma) [ (17*n^2 - 17*n + 2)/2 : n in [1..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%Y Cf. centered polygonal numbers listed in A069190.
%K easy,nice,nonn
%O 1,2
%A _Terrel Trotter, Jr._, Apr 07 2002
%E Typo in formula fixed by _Omar E. Pol_, Dec 22 2008
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