A000567 Octagonal numbers: n*(3*n-2). Also called star numbers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

From Floor van Lamoen, Jul 21 2001: (Start)

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,....

The spiral begins:

.

85--84--83--82--81--80

/ \

86 56--55--54--53--52 79

/ / \ \

87 57 33--32--31--30 51 78

/ / / \ \ \

88 58 34 16--15--14 29 50 77

/ / / / \ \ \ \

89 59 35 17 5---4 13 28 49 76

/ / / / / \ \ \ \ \

90 60 36 18 6 0 3 12 27 48 75

/ / / / / / / / / / /

91 61 37 19 7 1---2 11 26 47 74

\ \ \ \ \ . / / / /

92 62 38 20 8---9--10 25 46 73

\ \ \ \ . / / /

93 63 39 21--22--23--24 45 72

\ \ \ . / /

94 64 40--41--42--43--44 71

\ \ . /

95 65--66--67--68--69--70

\ .

96

.

(End)

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) = 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

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

The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016

For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-4; {1, 2n-2, 3, 2n-2, 1, 18n-8}]. For n=1, this collapses to [5; {5, 10}]. - Magus K. Chu, Oct 10 2022

a(n)*a(n+1) + 1 = (3n^2 + n - 1)^2. In general, a(n)*a(n+k) + k^2 = (3n^2 + (3k-2)n - k)^2. - Charlie Marion, May 23 2023

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
• 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.
• 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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 19-20.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
• Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.
• Cesar Ceballos and Viviane Pons, The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals, arXiv:2309.14261 [math.CO], 2023. See p. 42.
• C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
• John Elias, Illustration: Hexagonal spiral grids based on generalized octagonal numbers; Illustration: Generalized Square-Octagonal Grids.
• John Elias, Illustration: Nesting Cube-frames; Nesting Cube Animation; Nesting-frames Decomposition; Factorial Compartmentalization.
• 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.
• Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 342.
• Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 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.
• Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
• Viktor Levandovskyy, Christoph Koutschan, and Oleksandr Motsak, On Two-generated Non-commutative Algebras Subject to the Affine Relation, arXiv:1108.1108 [cs.SC], 2011.
• 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
• Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567.
• Leo Tavares, Illustration: Square Rays.
• Leo Tavares, Illustration: Twin Rectangular Rays.
• Leo Tavares, Illustration: Star Rows.
• Leo Tavares, Illustration: Split Stars.
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
• Eric Weisstein's World of Mathematics, Octagonal Number.
• Eric Weisstein's World of Mathematics, Wiener Index.
• Index to sequences related to polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Cf. A005408 (the odd numbers).

Cf. A000290, A000217, A046092, A000384, A002378, A003154.

Programs

GAP

List([0..50], n -> n*(3*n-2)); # G. C. Greubel, Nov 15 2018

Haskell

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

Magma

[n*(3*n-2) : n in [0..50]]; // Wesley Ivan Hurt, Oct 10 2021

Maple

A000567 := proc(n)
    n*(3*n-2) ;
end proc:
seq(A000567(n), n=1..50) ;

Mathematica

Table[n (3 n - 2), {n, 0, 50}] (* Harvey P. Dale, May 06 2012 *)
Table[PolygonalNumber[RegularPolygon[8], n], {n, 0, 43}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[8, Range[0, 20]] (* Eric W. Weisstein, Sep 07 2017 *)
LinearRecurrence[{3, -3, 1}, {1, 8, 21}, {0, 20}] (* Eric W. Weisstein, Sep 07 2017 *)

PARI

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

PARI

vector(50, n, n--; n*(3*n-2)) \\ G. C. Greubel, Nov 15 2018

Python

# Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 6, y + 6
A000567 = aList()
print([next(A000567) for i in range(49)]) # Peter Luschny, Aug 04 2019

Python

[n*(3*n-2) for n in range(50)] # Gennady Eremin, Mar 10 2022

Sage

[n*(3*n-2) for n in range(50)] # G. C. Greubel, Nov 15 2018

Formula

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

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

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

G.f.: x*(1+5*x)/(1-x)^3. Simon Plouffe in his 1992 dissertation

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

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

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

Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + 9*log(3))/12 = 1.2774090575596367311949534921... . - Vaclav Kotesovec, Apr 27 2016

From Ilya Gutkovskiy, Jul 29 2016: (Start)

Inverse binomial transform of A084857.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(2*sqrt(3)) = A093766. (End)

a(n) = n * A016777(n-1) = A053755(n) - A000290(n+1). - Bruce J. Nicholson, Aug 10 2017

Product_{n>=2} (1 - 1/a(n)) = 3/4. - Amiram Eldar, Jan 21 2021

P(4k+4,n) = ((k+1)*n - k)^2 - (k*n - k)^2 where P(m,n) is the n-th m-gonal number (a generalization of the Apr 10 2013 formula, a(n) = (2*n-1)^2 - (n-1)^2). - Charlie Marion, Oct 07 2021

From Leo Tavares, Oct 31 2021: (Start)

a(n) = A000290(n) + 4*A000217(n-1). See Square Rays illustration.

a(n) = A000290(n) + A046092(n-1)

a(n) = A000384(n) + 2*A000217(n-1). See Twin Rectangular Rays illustration.

a(n) = A000384(n) + A002378(n-1)

a(n) = A003154(n) - A045944(n-1). See Star Rows illustration. (End)

Extensions

Incorrect example removed by Joerg Arndt, Mar 11 2010