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A069127
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Centered 14-gonal numbers.
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8
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1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201, 4551, 4915, 5293, 5685, 6091, 6511, 6945, 7393, 7855, 8331, 8821, 9325, 9843, 10375, 10921, 11481, 12055, 12643, 13245
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OFFSET
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1,2
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COMMENTS
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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
Centered tetradecagonal numbers or centered tetrakaidecagonal numbers. - Omar E. Pol, Oct 03 2011
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 7*n^2 - 7*n + 1.
a(n) = 14*n+a(n-1)-14 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+12*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
a(n) = A163756(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = a(-n+1) = A193053(2n-2) + A193053(2n-3). - Bruno Berselli, Oct 21 2011
Sum_{n>=1} 1/a(n) = Pi * tan(sqrt(3/7)*Pi/2) / sqrt(21). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 8*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 8/e - 1. (End)
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EXAMPLE
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a(5) = 141 because 7*5^2 - 7*5 + 1 = 175 - 35 + 1 = 141.
a(5) = 71 because 71 = (7*5^2 - 7*5 + 2)/2 = (175 - 35 + 2)/2 = 142/2.
From Bruno Berselli, Oct 27 2017: (Start)
1 = -(1) + (2).
15 = -(1+2) + (3+4+5+6).
43 = -(1+2+3) + (4+5+6+7+8+9+10).
85 = -(1+2+3+4) + (5+6+7+8+9+10+11+12+13+14).
141 = -(1+2+3+4+5) + (6+7+8+9+10+11+12+13+14+15+16+17+18). (End)
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MATHEMATICA
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FoldList[#1 + #2 &, 1, 14 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Accumulate[14*Range[0, 50]]+1 (* Harvey P. Dale, Apr 09 2012 *)
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PROG
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(PARI) a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015
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CROSSREFS
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Cf. A005448, A001844, A005891, A003215, A069099.
Sequence in context: A280232 A233302 A072119 * A137183 A173873 A124708
Adjacent sequences: A069124 A069125 A069126 * A069128 A069129 A069130
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Terrel Trotter, Jr., Apr 07 2002
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STATUS
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approved
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