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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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30
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Except for the first term, numbers n to the first diagonal to A162245 (13, 37, 73, 121,..). Numbers n such that 6*n+3 is a square (From Gary Detlefs and Vincenzo Librandi) [From Vincenzo Librandi, Aug 08 2010]
Matone: the power of the Hodge bundle in the Mumford isomorphism. This is prime for n = {2, 3, 4, 6, 8, 9, 10, 11, 13, 14, 19, 20, 21, 23, 24, 31, 32, 33, 34, 36, 37, 39, 42, 43, 44, 46, 47, 48, 52, ...} = A184899 [Jonathan Vos Post, Feb 1, 2011]
Odd numbers of the form [n^2/6]. [From 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
| 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
| T. D. Noe, Table of n, a(n) for n=1..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Star Number
Marco Matone, Roberto Volpato, Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula, Jan 31, 2011.
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to centered polygonal numbers
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FORMULA
| G.f.: x*(1+10*x+x^2)/(1-x)^3.
a(n)= 1 + (sum(12*n)). 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
Binomial transform of [1, 12, 12, 0, 0, 0,...]; Narayana transform (A001263) of [1, 12, 0, 0, 0,...]. - Gary W. Adamson, Dec 29 2007
a(n) = 12*(n-1)+a(n-1), (with a(1)=1). [From Vincenzo Librandi, Aug 08 2010]
a(n) = A049598(n-1) + 1. - Omar E. Pol, Oct 03 2011
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EXAMPLE
| Contribution from Omar E. Pol, Aug 21 2011: (Start)
1. Classic illustration of initial terms of the stars numbers:
.
. 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
.
. 1 13 37
.
2. Alternative illustration of initial terms using n-1 concentric hexagons around of 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
| A003154:=-(1+10*z+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| lst={}; Do[AppendTo[lst, LegendreP[2, n]], {n, 1, 10^3, 2}]; lst [From Vladimir Orlovsky, Sep 11 2008]
FoldList[#1 + #2 &, 1, 12 Range@ 45] (* RGWv *)
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PROG
| (sage) [6* bernoulli_polynomial(n, 2) for n in xrange(1, 44)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
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CROSSREFS
| Cf. A007588, A049598, A003215, A001263.
Cf. A056827 [From Juri-Stepan Gerasimov, Jul 27 2011]
Sequence in context: A049742 A113601 A158864 * A083577 A155285 A155262
Adjacent sequences: A003151 A003152 A003153 * A003155 A003156 A003157
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Michael Somos
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