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A003154 Centered 12-gonal numbers. Also star numbers: 6*n*(n-1) + 1.
(Formerly M4893)
36
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; text; internal format)
OFFSET

1,2

COMMENTS

Binomial transform of [1, 12, 12, 0, 0, 0,...]. Narayana transform (A001263) of [1, 12, 0, 0, 0,...]. - Gary W. Adamson, Dec 29 2007

Except for the first term, numbers n to the first diagonal to A162245 (13, 37, 73, 121,..). - Vincenzo Librandi, Sep 28 2009

Numbers n such that 6*a(n)+3 is a square. - Gary Detlefs and 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 01 2011

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

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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 entries for linear recurrences with constant coefficients, signature (3,-3,1).

Index entries for sequences related to centered polygonal numbers

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

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

EXAMPLE

From Omar E. Pol, Aug 21 2011: (Start)

1. Classic illustration of initial terms of the star 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 a central element:

.

.                                 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)

MAPLE

A003154:=-(1+10*z+z**2)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

MATHEMATICA

lst={}; Do[AppendTo[lst, LegendreP[2, n]], {n, 1, 10^3, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

FoldList[#1 + #2 &, 1, 12 Range@ 45] (* Robert G. Wilson v *)

LinearRecurrence[{3, -3, 1}, {1, 13, 37}, 50] (* Harvey P. Dale, Jul 18 2016 *)

PROG

(sage) [6* bernoulli_polynomial(n, 2) for n in xrange(1, 44)] /* Zerinvary Lajos, May 17 2009 */

(PARI) a(n)=6*n*(n-1)+1 \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A007588, A049598, A003215, A001263, A056827.

Sequence in context: A247867 A113601 A158864 * A083577 A155285 A155262

Adjacent sequences:  A003151 A003152 A003153 * A003155 A003156 A003157

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos

STATUS

approved

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Last modified September 26 01:05 EDT 2016. Contains 276541 sequences.