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A003154 Centered 12-gonal numbers, or centered dodecagonal numbers: numbers of the form 6*k*(k-1) + 1.
(Formerly M4893)
79
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 (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
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
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
M. Gardner and N. J. A. Sloane, Correspondence, 1973-74
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).
Eric Weisstein's World of Mathematics, Star Number
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
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:=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 *)
PROG
(PARI) a(n)=6*n*(n-1)+1 \\ Charles R Greathouse IV, Nov 20 2012
(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
(GAP) List([1..50], n-> 12*Binomial(n, 2)+1 ); # G. C. Greubel, Jul 23 2019
(Python)
print([6*n*(n-1)+1 for n in range(1, 47)]) # Michael S. Branicky, Jan 13 2021
CROSSREFS
Row 4 of A257565.
Sequence in context: A247867 A113601 A158864 * A083577 A155285 A155262
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Michael Somos
STATUS
approved

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Last modified February 27 21:03 EST 2024. Contains 370378 sequences. (Running on oeis4.)