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%I M4893 #190 Dec 21 2023 10:34:04
%S 1,13,37,73,121,181,253,337,433,541,661,793,937,1093,1261,1441,1633,
%T 1837,2053,2281,2521,2773,3037,3313,3601,3901,4213,4537,4873,5221,
%U 5581,5953,6337,6733,7141,7561,7993,8437,8893,9361,9841,10333,10837,11353,11881,12421
%N Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.
%C Binomial transform of [1, 12, 12, 0, 0, 0, ...]. Narayana transform (A001263) of [1, 12, 0, 0, 0, ...]. - _Gary W. Adamson_, Dec 29 2007
%C Numbers k such that 6*k+3 is a square, these squares are given in A016946. - _Gary Detlefs_ and _Vincenzo Librandi_, Aug 08 2010
%C Odd numbers of the form floor(n^2/6). - _Juri-Stepan Gerasimov_, Jul 27 2011
%C Bisection of A032528. - _Omar E. Pol_, Aug 20 2011
%C 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
%C The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered triangular numbers A005448(n). - _Peter M. Chema_, Dec 20 2023
%D M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003154/b003154.txt">Table of n, a(n) for n = 1..1000</a>
%H John Elias, <a href="/A003154/a003154.png">Illustration: Star Configurations on the Zero-Centered Hexagonal Number Spiral</a>
%H John Elias, <a href="/A003154/a003154_1.png">Illustration: Star Configurations on the Zero-Centered Square and Hexagonal Number Spirals</a>
%H John Elias, <a href="/A003154/a003154_2.png">Illustration: Generalized Pentagonal and Octagonal Numbers in the Star-Crossed Configurations</a>
%H John Elias, <a href="/A003154/a003154_3.png">Illustration: Generalized Pentagonal and Octagonal Integration in Centered 9-gonal Triangles</a>
%H M. Gardner and N. J. A. Sloane, <a href="/A003154/a003154.pdf">Correspondence, 1973-74</a>
%H Marco Matone and Roberto Volpato, <a href="http://arxiv.org/abs/1102.0006">Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula</a>, arXiv:1102.0006 [math.AG], 2011-2012, c_n.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Amelia C. Sparavigna, <a href="https://iris.polito.it/retrieve/handle/11583/2750477/269564/Starnumbers.pdf">Groupoid of OEIS A003154 numbers (star numbers or centered dodecagonal numbers</a>, Politecnico di Torino, Repository istituzionale (2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3387054">Groupoid of OEIS A003154 Numbers (star numbers or centered dodecagonal numbers)</a>, Department of Applied Science and Technology, Politecnico di Torino (Italy, 2019).
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.4662348">Generalized Sum of Stella Octangula Numbers</a>, Politecnico di Torino (Italy, 2021).
%H Leo Tavares, <a href="/A003154/a003154.jpg">Illustration: Twin Hexagons</a>
%H Leo Tavares, <a href="/A003154/a003154_1.jpg">Illustration: Diamond Rays</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarNumber.html">Star Number</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%F G.f.: x*(1+10*x+x^2)/(1-x)^3. _Simon Plouffe_ in his 1992 dissertation
%F 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
%F 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
%F a(n) = 12*(n-1) + a(n-1), with n>1, a(1)=1. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = A049598(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F Sum_{n>=1} 1/a(n) = A306980 = Pi * tan(Pi/(2*sqrt(3))) / (2*sqrt(3)). - _Vaclav Kotesovec_, Jul 23 2019
%F From _Amiram Eldar_, Jun 21 2020: (Start)
%F Sum_{n>=1} a(n)/n! = 7*e - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 7/e - 1. (End)
%F a(n) = 2*A003215(n-1) - 1. - _Leo Tavares_, Jul 30 2021
%F E.g.f.: exp(x)*(1 + 6*x^2) - 1. - _Stefano Spezia_, Aug 19 2022
%e From _Omar E. Pol_, Aug 21 2011: (Start)
%e 1. Classic illustration of initial terms of the star numbers:
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%e 2. Alternative illustration of initial terms using n-1 concentric hexagons around a central element:
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%e (End)
%p A003154:=n->6*n*(n-1) + 1: seq(A003154(n), n=1..100); # _Wesley Ivan Hurt_, Oct 23 2017
%t FoldList[#1 + #2 &, 1, 12 Range@50] (* _Robert G. Wilson v_ *)
%t LinearRecurrence[{3,-3,1},{1,13,37},50] (* _Harvey P. Dale_, Jul 18 2016 *)
%t 12*Binomial[Range[50], 2] + 1 (* _G. C. Greubel_, Jul 23 2019 *)
%o (PARI) a(n)=6*n*(n-1)+1 \\ _Charles R Greathouse IV_, Nov 20 2012
%o (J) ([: >: 6 * ] * <:) i.1000 NB. _Stephen Makdisi_, May 06 2018
%o (Magma) [12*Binomial(n,2)+1: n in [1..50]]; // _G. C. Greubel_, Jul 23 2019
%o (GAP) List([1..50], n-> 12*Binomial(n,2)+1 ); # _G. C. Greubel_, Jul 23 2019
%o (Python)
%o print([6*n*(n-1)+1 for n in range(1, 47)]) # _Michael S. Branicky_, Jan 13 2021
%Y Cf. A001263, A001318, A003215, A007588, A016946, A032528, A033581, A049598, A056827, A306980.
%Y Row 4 of A257565.
%Y Cf. A000217, A005448, A016754, A146325.
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_
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