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A069127 Centered 14-gonal numbers. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
FORMULA
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)
a(n) = A069099(n) + 7*A000217(n-1). - Leo Tavares, Jul 09 2021
EXAMPLE
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)
MATHEMATICA
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 *)
PROG
(PARI) a(n)=7*n^2-7*n+1 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Sequence in context: A280232 A233302 A072119 * A137183 A173873 A124708
KEYWORD
nonn,easy,nice
AUTHOR
Terrel Trotter, Jr., Apr 07 2002
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)