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%I #61 Jul 14 2021 14:56:48
%S 1,15,43,85,141,211,295,393,505,631,771,925,1093,1275,1471,1681,1905,
%T 2143,2395,2661,2941,3235,3543,3865,4201,4551,4915,5293,5685,6091,
%U 6511,6945,7393,7855,8331,8821,9325,9843,10375,10921,11481,12055,12643,13245
%N Centered 14-gonal numbers.
%C Binomial transform of [1, 14, 14, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 14, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 29 2011
%C Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - _Omar E. Pol_, Oct 03 2011
%H T. D. Noe, <a href="/A069127/b069127.txt">Table of n, a(n) for n = 1..1000</a>
%H Leo Tavares, <a href="/A069127/a069127.jpg">Illustration: Heptagonal Stars</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) = 7*n^2 - 7*n + 1.
%F a(n) = 14*n+a(n-1)-14 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F G.f.: -x*(1+12*x+x^2) / (x-1)^3. - _R. J. Mathar_, Feb 04 2011
%F a(n) = A163756(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - _Bruno Berselli_, Oct 21 2011
%F Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - _Vaclav Kotesovec_, Jul 23 2019
%F From _Amiram Eldar_, Jun 21 2020: (Start)
%F Sum_{n>=1} a(n)/n! = 8*e - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)
%F a(n) = A069099(n) + 7*A000217(n-1). - _Leo Tavares_, Jul 09 2021
%e a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
%e a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
%e From _Bruno Berselli_, Oct 27 2017: (Start)
%e 1 = -(1) + (2).
%e 15 = -(1+2) + (3+4+5+6).
%e 43 = -(1+2+3) + (4+5+6+7+8+9+10).
%e 85 = -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
%e 141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)
%t FoldList[#1 + #2 &, 1, 14 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t Accumulate[14*Range[0,50]]+1 (* _Harvey P. Dale_, Apr 09 2012 *)
%o (PARI) a(n)=7*n^2-7*n+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A005448, A001844, A005891, A003215, A069099.
%Y Cf. A163756, A193053.
%K nonn,easy,nice
%O 1,2
%A _Terrel Trotter, Jr._, Apr 07 2002
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