|
| |
|
|
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
|
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).
|
|
|
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
|
|
|
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
|
Cf. A005448, A001844, A005891, A003215, A069099.
Sequence in context: A280232 A233302 A072119 * A137183 A173873 A124708
Adjacent sequences: A069124 A069125 A069126 * A069128 A069129 A069130
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
Terrel Trotter, Jr., Apr 07 2002
|
|
|
STATUS
|
approved
|
| |
|
|
