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A003154
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Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1.
(Formerly M4893)
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70
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1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837, 11353, 11881, 12421
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Binomial transform of [1, 12, 12, 0, 0, 0, ...]. Narayana transform (A001263) of [1, 12, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
Numbers k such that 6*k+3 is a square, these squares are given in A016946. - Gary Detlefs and Vincenzo Librandi, Aug 08 2010
Odd numbers of the form floor(n^2/6). - Juri-Stepan Gerasimov, Jul 27 2011
Bisection of A032528. - Omar E. Pol, Aug 20 2011
Sequence found by reading the line from 1, in the direction 1, 13, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033581 in the same spiral. - Omar E. Pol, Sep 08 2011
Centered dodecagonal numbers. - Omar E. Pol, Oct 03 2011
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REFERENCES
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M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
M. Gardner and N. J. A. Sloane, Correspondence, 1973-74
Marco Matone and Roberto Volpato, Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula, arXiv:1102.0006 [math.AG], 2011-2012, c_n.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Amelia C. Sparavigna, Groupoid of OEIS A003154 numbers (star numbers or centered dodecagonal numbers, Politecnico di Torino, Repository istituzionale (2019).
Amelia Carolina Sparavigna, Groupoid of OEIS A003154 Numbers (star numbers or centered dodecagonal numbers), Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Leo Tavares, Illustration: Twin Hexagons
Eric Weisstein's World of Mathematics, Star Number
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to centered polygonal numbers
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FORMULA
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G.f.: x*(1+10*x+x^2)/(1-x)^3. Simon Plouffe in his 1992 dissertation
a(n) = 1 + Sum_{j=0..n} (12*j). E.g., a(2)=37 because 1 + 12*0 + 12*1 + 12*2 = 37. - _Xavier Acloque_, Oct 06 2003
a(n) = numerator in B_2(x) = (1/2)x^2 - (1/2)x + 1/12 = Bernoulli polynomial of degree 2. - Gary W. Adamson, May 30 2005
a(n) = 12*(n-1) + a(n-1), with n>1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = A049598(n-1) + 1. - Omar E. Pol, Oct 03 2011
Sum_{n>=1} 1/a(n) = A306980 = Pi * tan(Pi/(2*sqrt(3))) / (2*sqrt(3)). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 7*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 7/e - 1. (End)
a(n) = 2*A003215(n-1) - 1. - Leo Tavares, Jul 30 2021
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EXAMPLE
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From Omar E. Pol, Aug 21 2011: (Start)
1. Classic illustration of initial terms of the star numbers:
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. o
. o o
. o o o o o o o o
. o o o o o o o o o o
. o o o o o o o o o
. o o o o o o o o o o
. o o o o o o o o
. o o
. o
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. 1 13 37
.
2. Alternative illustration of initial terms using n-1 concentric hexagons around a central element:
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. o o o o o
. o o
. o o o o o o o o
. o o o o o o
. o o o o o o o o o
. o o o o o o
. o o o o o o o o
. o o
. o o o o o
(End)
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MAPLE
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A003154:=n->6*n*(n-1) + 1: seq(A003154(n), n=1..100); # Wesley Ivan Hurt, Oct 23 2017
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MATHEMATICA
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FoldList[#1 + #2 &, 1, 12 Range@50] (* Robert G. Wilson v *)
LinearRecurrence[{3, -3, 1}, {1, 13, 37}, 50] (* Harvey P. Dale, Jul 18 2016 *)
12*Binomial[Range[50], 2] + 1 (* G. C. Greubel, Jul 23 2019 *)
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PROG
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(PARI) a(n)=6*n*(n-1)+1 \\ Charles R Greathouse IV, Nov 20 2012
(J) ([: >: 6 * ] * <:) i.1000 NB. Stephen Makdisi, May 06 2018
(MAGMA) [12*Binomial(n, 2)+1: n in [1..50]]; // G. C. Greubel, Jul 23 2019
(GAP) List([1..50], n-> 12*Binomial(n, 2)+1 ); # G. C. Greubel, Jul 23 2019
(Python)
print([6*n*(n-1)+1 for n in range(1, 47)]) # Michael S. Branicky, Jan 13 2021
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CROSSREFS
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Cf. A001263, A003215, A007588, A049598, A056827, A306980. Row 4 of A257565.
Sequence in context: A247867 A113601 A158864 * A083577 A155285 A155262
Adjacent sequences: A003151 A003152 A003153 * A003155 A003156 A003157
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Michael Somos
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STATUS
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approved
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