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%I A003154 M4893 #203 Aug 01 2024 11:50:01
%S A003154 1,13,37,73,121,181,253,337,433,541,661,793,937,1093,1261,1441,1633,
%T A003154 1837,2053,2281,2521,2773,3037,3313,3601,3901,4213,4537,4873,5221,
%U A003154 5581,5953,6337,6733,7141,7561,7993,8437,8893,9361,9841,10333,10837,11353,11881,12421
%N A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.
%C A003154 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 A003154 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 A003154 Odd numbers of the form floor(n^2/6). - _Juri-Stepan Gerasimov_, Jul 27 2011
%C A003154 Bisection of A032528. - _Omar E. Pol_, Aug 20 2011
%C A003154 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 A003154 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 A003154 Martin Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 20.
%D A003154 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003154 T. D. Noe, <a href="/A003154/b003154.txt">Table of n, a(n) for n = 1..1000</a>
%H A003154 John Elias, <a href="/A003154/a003154.png">Illustration: Star Configurations on the Zero-Centered Hexagonal Number Spiral</a>.
%H A003154 John Elias, <a href="/A003154/a003154_1.png">Illustration: Star Configurations on the Zero-Centered Square and Hexagonal Number Spirals</a>.
%H A003154 John Elias, <a href="/A003154/a003154_2.png">Illustration: Generalized Pentagonal and Octagonal Numbers in the Star-Crossed Configurations</a>.
%H A003154 John Elias, <a href="/A003154/a003154_3.png">Illustration: Generalized Pentagonal and Octagonal Integration in Centered 9-gonal Triangles</a>.
%H A003154 Martin Gardner and N. J. A. Sloane, <a href="/A003154/a003154.pdf">Correspondence, 1973-74</a>.
%H A003154 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 A003154 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 A003154 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A003154 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 A003154 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 A003154 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 A003154 Leo Tavares, <a href="/A003154/a003154.jpg">Illustration: Twin Hexagons</a>.
%H A003154 Leo Tavares, <a href="/A003154/a003154_1.jpg">Illustration: Diamond Rays</a>.
%H A003154 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StarNumber.html">Star Number</a>.
%H A003154 R. Yin, J. Mu, and T. Komatsu, <a href="https://doi.org/10.20944/preprints202407.2280.v1">The p-Frobenius Number for the Triple of the Generalized Star Numbers</a>, Preprints 2024, 2024072280. See p. 1.
%H A003154 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H A003154 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.
%F A003154 G.f.: x*(1+10*x+x^2)/(1-x)^3. _Simon Plouffe_ in his 1992 dissertation
%F A003154 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 A003154 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 A003154 a(n) = 12*(n-1) + a(n-1), with n>1, a(1)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A003154 a(n) = A049598(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F A003154 Sum_{n>=1} 1/a(n) = A306980 = Pi * tan(Pi/(2*sqrt(3))) / (2*sqrt(3)). - _Vaclav Kotesovec_, Jul 23 2019
%F A003154 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A003154 Sum_{n>=1} a(n)/n! = 7*e - 1.
%F A003154 Sum_{n>=1} (-1)^n * a(n)/n! = 7/e - 1. (End)
%F A003154 a(n) = 2*A003215(n-1) - 1. - _Leo Tavares_, Jul 30 2021
%F A003154 E.g.f.: exp(x)*(1 + 6*x^2) - 1. - _Stefano Spezia_, Aug 19 2022
%e A003154 From _Omar E. Pol_, Aug 21 2011: (Start)
%e A003154 1. Classic illustration of initial terms of the star numbers:
%e A003154 .
%e A003154 .                                     o
%e A003154 .                                    o o
%e A003154 .                  o            o o o o o o o
%e A003154 .               o o o o          o o o o o o
%e A003154 .     o          o o o            o o o o o
%e A003154 .               o o o o          o o o o o o
%e A003154 .                  o            o o o o o o o
%e A003154 .                                    o o
%e A003154 .                                     o
%e A003154 .
%e A003154 .     1            13                 37
%e A003154 .
%e A003154 2. Alternative illustration of initial terms using n-1 concentric hexagons around a central element:
%e A003154 .
%e A003154 .                                 o o o o o
%e A003154 .                                o         o
%e A003154 .                o o o          o   o o o   o
%e A003154 .               o     o        o   o     o   o
%e A003154 .     o        o   o   o      o   o   o   o   o
%e A003154 .               o     o        o   o     o   o
%e A003154 .                o o o          o   o o o   o
%e A003154 .                                o         o
%e A003154 .                                 o o o o o
%e A003154 (End)
%p A003154 A003154:=n->6*n*(n-1) + 1: seq(A003154(n), n=1..100); # _Wesley Ivan Hurt_, Oct 23 2017
%t A003154 FoldList[#1 + #2 &, 1, 12 Range@50] (* _Robert G. Wilson v_ *)
%t A003154 LinearRecurrence[{3,-3,1},{1,13,37},50] (* _Harvey P. Dale_, Jul 18 2016 *)
%t A003154 12*Binomial[Range[50], 2] + 1 (* _G. C. Greubel_, Jul 23 2019 *)
%o A003154 (PARI) a(n)=6*n*(n-1)+1 \\ _Charles R Greathouse IV_, Nov 20 2012
%o A003154 (J) ([: >: 6 * ] * <:) i.1000 NB. _Stephen Makdisi_, May 06 2018
%o A003154 (Magma) [12*Binomial(n,2)+1: n in [1..50]]; // _G. C. Greubel_, Jul 23 2019
%o A003154 (GAP) List([1..50], n-> 12*Binomial(n,2)+1 ); # _G. C. Greubel_, Jul 23 2019
%o A003154 (Python)
%o A003154 print([6*n*(n-1)+1 for n in range(1, 47)]) # _Michael S. Branicky_, Jan 13 2021
%Y A003154 Cf. A001263, A001318, A003215, A007588, A016946, A032528, A033581, A049598, A056827, A306980.
%Y A003154 Row 4 of A257565.
%Y A003154 Cf. A000217, A005448, A016754, A146325.
%K A003154 nonn,easy,nice
%O A003154 1,2
%A A003154 _N. J. A. Sloane_
%E A003154 More terms from _Michael Somos_

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