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%I A069127 #68 Aug 01 2024 11:58:14
%S A069127 1,15,43,85,141,211,295,393,505,631,771,925,1093,1275,1471,1681,1905,
%T A069127 2143,2395,2661,2941,3235,3543,3865,4201,4551,4915,5293,5685,6091,
%U A069127 6511,6945,7393,7855,8331,8821,9325,9843,10375,10921,11481,12055,12643,13245,13861,14491
%N A069127 Centered 14-gonal numbers.
%C A069127 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 A069127 Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - _Omar E. Pol_, Oct 03 2011
%H A069127 T. D. Noe, <a href="/A069127/b069127.txt">Table of n, a(n) for n = 1..1000</a>
%H A069127 Leo Tavares, <a href="/A069127/a069127.jpg">Illustration: Heptagonal Stars</a>.
%H A069127 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.
%H A069127 R. Yin, J. Mu, and T. Komatsu, <a href="https://doi.org/10.20944/preprints202407.2280.v1">The p-Frobenius Number for the Triple of the Generalized Star Numbers</a>, Preprints 2024, 2024072280. See p. 2.
%H A069127 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.
%H A069127 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A069127 a(n) = 7*n^2 - 7*n + 1.
%F A069127 a(n) = 14*n+a(n-1)-14 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F A069127 G.f.: -x*(1+12*x+x^2) / (x-1)^3. - _R. J. Mathar_, Feb 04 2011
%F A069127 a(n) = A163756(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F A069127 a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - _Bruno Berselli_, Oct 21 2011
%F A069127 Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - _Vaclav Kotesovec_, Jul 23 2019
%F A069127 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A069127 Sum_{n>=1} a(n)/n! = 8*e - 1.
%F A069127 Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)
%F A069127 a(n) = A069099(n) + 7*A000217(n-1). - _Leo Tavares_, Jul 09 2021
%F A069127 E.g.f.: exp(x)*(1 + 7*x^2) - 1. - _Stefano Spezia_, Aug 01 2024
%e A069127 a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
%e A069127 a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
%e A069127 From _Bruno Berselli_, Oct 27 2017: (Start)
%e A069127 1   =         -(1) + (2).
%e A069127 15  =       -(1+2) + (3+4+5+6).
%e A069127 43  =     -(1+2+3) + (4+5+6+7+8+9+10).
%e A069127 85  =   -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
%e A069127 141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)
%t A069127 FoldList[#1 + #2 &, 1, 14 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A069127 Accumulate[14*Range[0,50]]+1 (* _Harvey P. Dale_, Apr 09 2012 *)
%o A069127 (PARI) a(n)=7*n^2-7*n+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A069127 Cf. A005448, A001844, A005891, A003215, A069099.
%Y A069127 Cf. A163756, A193053.
%K A069127 nonn,easy,nice
%O A069127 1,2
%A A069127 _Terrel Trotter, Jr._, Apr 07 2002

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