login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000567 Octagonal numbers: n*(3*n-2). Also called star numbers.
(Formerly M4493 N1901)
146
0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201, 3400, 3605, 3816, 4033, 4256, 4485, 4720, 4961, 5208, 5461 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,1,... - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jul 21 2001. The spiral begins:

......16..15..14

....17..5...4...13

..18..6...0...3...12

19..7...1...2...11..26

..20..8...9...10..25

....21..22..23..24

a(n) = (3n-2)(3n-1)(3n)/{(3n-1)+(3n-2)+(3n)} i.e. the product of three consecutive numbers/their sum. a(1) = 1*2*3/(1+2+3),a(2) = 4*5*6/(4+5+6), etc. - Amarnath Murthy, Aug 29 2002

From Lekraj Beedassy, Oct 02 2003: (Start)

Also the number of distinct three-cell blocks that may be removed out of A000217(n+1) square cells arranged in a stepping triangular array of side (n+1). A 5-layer triangular array of square cells, for instance, has vertices outlined thus:

x x

x x x

x x x x

x x x x x

x x x x x x

x x x x x x  (End)

First derivative at n of A045991 - Ross La Haye, Oct 23 2004

Starting from n=1, the sequence corresponds to the Wiener index of K_{n,n} (the complete bipartite graph wherein each independent set has n vertices). - Kailasam Viswanathan Iyer, Mar 11 2009

Number of divisors of 24^(n-1) for n>0 (cf A009968). - J. Lowell, Aug 30 2008

a(n) = A000578(n) - A007531(n). [Reinhard Zumkeller, Sep 18 2009]

a(n) = A001399(6n-5), number of partitions of 6*n-5 into parts <4. For example a(2)=8 and partitions of 6*2-5=7 into parts <4 are: [1,1,1,1,1,1,1], [1,1,1,1,1,2],[1,1,1,1,3], [1,1,1,2,2], [1,1,2,3], [1,2,2,2], [1,3,3], [2,2,3]. - Adi Dani, Jun 07 2011

Also, sequence found by reading the line from 0 in the direction 0, 8,..., and the parallel line from 1 in the direction 1, 21,..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011

a(n) = (A185212(n) - 1) / 4. - Reinhard Zumkeller, Dec 20 2012

Partial sums give A002414. - Omar E. Pol, Jan 12 2013

Generate a Pythagorean triple using Euclid's formula with (n, n-1) to give A,B,C. a(n) = B + (A + C)/2. - J. M. Bergot, Jul 13 2013

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 1.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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 = 0..1000

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 342

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013

R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Table 1.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Viktor Levandovskyy, Christoph Koutschan and Oleksandr Motsak, On Two-generated Non-commutative Algebras Subject to the Affine Relation, arXiv:1108.1108, 2011.

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.

Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567

Eric Weisstein's World of Mathematics, Octagonal Number

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = n*(3*n-2).

E.g.f. : exp(x)*(x+3*x^2). - Paul Barry, Jul 23 2003

G.f.: x*(1+5*x)/(1-x)^3.

a(n) = sum{k=1..n, 5*n-4*k} - Paul Barry, Sep 06 2005

a(n) = n+6*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 14 2005

a(n) = C(n+1,2) + 5*C(n,2).

Starting (1, 8, 21, 40, 65,...) = binomial transform of [1, 7, 6, 0, 0, 0,...]. - Gary W. Adamson, Apr 30 2008

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=8. [Jaume Oliver Lafont, Dec 02 2008]

a(n) = a(n-1) + 6*n - 5 (with a(0)=0). [Vincenzo Librandi, Nov 20 2010]

a(n) = 2*a(n-1) - a(n-2) + 6. [Ant King, Sep 01 2011]

a(n) = A000217(n)+5*A000217(n-1). [Vincenzo Librandi, Nov 20 2010]

a(n) = A174709(6n). - Philippe Deléham, Mar 26 2013

a(n) = (2*n-1)^2 - (n-1)^2. - Ivan N. Ianakiev, Apr 10 2013

a(6*a(n)+16*n+1) = a(6*a(n)+16*n) + a(6*n+1). - Vladimir Shevelev, Jan 24 2014

a(0) = 0, a(n) = sum_{k=0..n-1} A005408(A051162(n-1,k)), n>=1. - L. Edson Jeffery, Jul 28 2014

MAPLE

[ seq(n*(3*n-2), n=1..50) ];

A000567:=-(1+5*z)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+6 od: seq(a[n], n=0..43); - Zerinvary Lajos, Feb 18 2008

MATHEMATICA

Table[n(3n-2), {n, 0, 50}] (* Harvey P. Dale, May 06 2012 *)

PROG

(PARI) a(n)=n*(3*n-2) \\ Charles R Greathouse IV, Jun 10 2011

(Haskell)

a000567 n = n * (3 * n - 2)  -- Reinhard Zumkeller, Dec 20 2012

CROSSREFS

Cf. A014641, A014642, A014793, A014794, A001835, A016777, A045944, A093563 ((6, 1) Pascal, column m=2). A016921 (differences).

n-gonal numbers: A000217, A000290, A000326, A000566, A001106, A001107, A051682, A051624, A051865-A051876.

Cf. A005408 (the odd numbers).

Sequence in context: A179681 A224039 A225287 * A124484 A137742 A190456

Adjacent sequences:  A000564 A000565 A000566 * A000568 A000569 A000570

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Removed incorrect example Joerg Arndt, Mar 11 2010

Broken link to Hyun Kwang Kim's paper fixed by Felix Fröhlich, Jun 16 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 1 03:38 EDT 2014. Contains 245104 sequences.