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%I #61 Feb 07 2023 08:38:00
%S 1,16,46,91,151,226,316,421,541,676,826,991,1171,1366,1576,1801,2041,
%T 2296,2566,2851,3151,3466,3796,4141,4501,4876,5266,5671,6091,6526,
%U 6976,7441,7921,8416,8926,9451,9991,10546,11116,11701,12301,12916,13546,14191,14851,15526
%N Centered 15-gonal numbers: a(n) = (15*n^2 - 15*n + 2)/2.
%C Centered pentadecagonal numbers or centered quindecagonal numbers or centered pentakaidecagonal numbers. - _Omar E. Pol_, Oct 03 2011
%H T. D. Noe, <a href="/A069128/b069128.txt">Table of n, a(n) for n = 1..1000</a>
%H E. Weisstein, <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) = (15*n^2 - 15*n + 2)/2.
%F a(n) = 15*n+a(n-1)-15 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F G.f.: -x*(1+13*x+x^2) / (x-1)^3. - _R. J. Mathar_, Feb 04 2011
%F Binomial transform of [1, 15, 15, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 15, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 28 2011
%F a(n) = A194715(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F From _Amiram Eldar_, Jun 21 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(7/15)*Pi/2)/sqrt(105).
%F Sum_{n>=1} a(n)/n! = 17*e/2 - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 17/(2*e) - 1. (End)
%F E.g.f.: exp(x)*(1 + 15*x^2/2) - 1. - _Nikolaos Pantelidis_, Feb 07 2023
%e a(5) = 151 because (15*5^2 - 15*5 + 2)/2 = 151.
%p A069128:=n->(15*n^2 - 15*n + 2)/2: seq(A069128(n), n=1..50); # _Wesley Ivan Hurt_, Nov 14 2014
%t FoldList[#1 + #2 &, 1, 15 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t LinearRecurrence[{3,-3,1},{1,16,46},50] (* _Harvey P. Dale_, Oct 22 2013 *)
%o (Magma) [(15*n^2 - 15*n + 2)/2 : n in [1..50]]; // _Wesley Ivan Hurt_, Nov 14 2014
%o (PARI) a(n)=15*n*(n-1)/2+1 \\ _Charles R Greathouse IV_, Nov 15 2014
%Y Cf. A005448, A001844, A005891, A003215, A069099.
%K nonn,easy,nice
%O 1,2
%A _Terrel Trotter, Jr._, Apr 07 2002
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