A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

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

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

Examples

			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)
		

References

• Martin 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
• John Elias, Illustration: Star Configurations on the Zero-Centered Hexagonal Number Spiral.
• John Elias, Illustration: Star Configurations on the Zero-Centered Square and Hexagonal Number Spirals.
• John Elias, Illustration: Generalized Pentagonal and Octagonal Numbers in the Star-Crossed Configurations.
• John Elias, Illustration: Generalized Pentagonal and Octagonal Integration in Centered 9-gonal Triangles.
• Martin 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.
• Leo Tavares, Illustration: Diamond Rays.
• Eric Weisstein's World of Mathematics, Star Number.
• R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 1.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
• Index entries for sequences related to centered polygonal numbers.

Crossrefs

Cf. A001263, A001318, A003215, A007588, A016946, A032528, A033581, A049598, A056827, A306980.

Row 4 of A257565.

Cf. A000217, A005448, A016754, A146325.

Programs

GAP

List([1..50], n-> 12*Binomial(n,2)+1 ); # G. C. Greubel, Jul 23 2019

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

Maple

A003154:=n->6*n*(n-1) + 1: seq(A003154(n), n=1..100); # Wesley Ivan Hurt, Oct 23 2017

Mathematica

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 *)

PARI

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

Python

print([6*n*(n-1)+1 for n in range(1, 47)]) # Michael S. Branicky, Jan 13 2021

Formula

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

E.g.f.: exp(x)*(1 + 6*x^2) - 1. - Stefano Spezia, Aug 19 2022

Extensions

More terms from Michael Somos