%I M4493 N1901 #335 Sep 07 2024 15:43:51

%S 0,1,8,21,40,65,96,133,176,225,280,341,408,481,560,645,736,833,936,

%T 1045,1160,1281,1408,1541,1680,1825,1976,2133,2296,2465,2640,2821,

%U 3008,3201,3400,3605,3816,4033,4256,4485,4720,4961,5208,5461

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

%C From _Floor van Lamoen_, Jul 21 2001: (Start)

%C 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,....

%C The spiral begins:

%C .

%C 85--84--83--82--81--80

%C / \

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

%C / / \ \

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

%C / / / \ \ \

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

%C / / / / \ \ \ \

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

%C / / / / / \ \ \ \ \

%C 90 60 36 18 6 0 3 12 27 48 75

%C / / / / / / / / / / /

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

%C \ \ \ \ \ . / / / /

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

%C \ \ \ \ . / / /

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

%C \ \ \ . / /

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

%C \ \ . /

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

%C \ .

%C 96

%C .

%C From _Lekraj Beedassy_, Oct 02 2003: (Start)

%C 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:

%C x x

%C x x x

%C x x x x

%C x x x x x

%C x x x x x x

%C x x x x x x (End)

%C First derivative at n of A045991. - _Ross La Haye_, Oct 23 2004

%C 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

%C Number of divisors of 24^(n-1) for n > 0 (cf A009968). - _J. Lowell_, Aug 30 2008

%C 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

%C 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

%C Partial sums give A002414. - _Omar E. Pol_, Jan 12 2013

%C 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

%C 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

%C 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

%C 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

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

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

%D 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.

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

%H T. D. Noe, <a href="/A000567/b000567.txt">Table of n, a(n) for n = 0..1000</a>

%H Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, <a href="https://arxiv.org/abs/2403.10500">A lozenge triangulation of the plane with integers</a>, arXiv:2403.10500 [math.NT], 2024.

%H Francesco Brenti and Paolo Sentinelli, <a href="https://arxiv.org/abs/2212.04932">Wachs permutations, Bruhat order and weak order</a>, arXiv:2212.04932 [math.CO], 2022.

%H Cesar Ceballos and Viviane Pons, <a href="https://arxiv.org/abs/2309.14261">The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals</a>, arXiv:2309.14261 [math.CO], 2023. See p. 42.

%H C. K. Cook and M. R. Bacon, <a href="https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343.

%H John Elias, <a href="/A000567/a000567.png">Illustration: Hexagonal spiral grids based on generalized octagonal numbers</a>; <a href="/A000567/a000567_1.png">Illustration: Generalized Square-Octagonal Grids</a>.

%H John Elias, <a href="/A000567/a000567_2.png">Illustration: Nesting Cube-frames</a>; <a href="/A000567/a000567.gif">Nesting Cube Animation</a>; <a href="/A000567/a000567_3.png">Nesting-frames Decomposition</a>; <a href="/A000567/a000567_4.png">Factorial Compartmentalization</a>.

%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

%H Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=342">Encyclopedia of Combinatorial Structures 342</a>.

%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.

%H R. Kemp, <a href="http://dx.doi.org/10.1016/0012-365X(82)90123-6">On the number of words in the language {w in Sigma* | w = w^R }^2</a>, Discrete Math., 40 (1982), 225-234. See Table 1.

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.

%H Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, <a href="https://arxiv.org/abs/2010.15636">Exact solutions in low-rank approximation with zeros</a>, arXiv:2010.15636 [math.AG], 2020.

%H Viktor Levandovskyy, Christoph Koutschan, and Oleksandr Motsak, <a href="http://arxiv.org/abs/1108.1108">On Two-generated Non-commutative Algebras Subject to the Affine Relation</a>, arXiv:1108.1108 [cs.SC], 2011.

%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567</a>.

%H Leo Tavares, <a href="/A000567/a000567.jpg">Illustration: Square Rays</a>.

%H Leo Tavares, <a href="/A000567/a000567_2.jpg">Illustration: Twin Rectangular Rays</a>.

%H Leo Tavares, <a href="/A000567/a000567_1.jpg">Illustration: Star Rows</a>.

%H Leo Tavares, <a href="/A000567/a000567_3.jpg">Illustration: Split Stars</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F 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

%F E.g.f.: exp(x)*(x+3*x^2). - _Paul Barry_, Jul 23 2003

%F G.f.: x*(1+5*x)/(1-x)^3. _Simon Plouffe_ in his 1992 dissertation

%F a(n) = Sum_{k=1..n} (5*n - 4*k). - _Paul Barry_, Sep 06 2005

%F a(n) = n + 6*A000217(n-1). - _Floor van Lamoen_, Oct 14 2005

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

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

%F 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

%F a(n) = A000578(n) - A007531(n). - _Reinhard Zumkeller_, Sep 18 2009

%F a(n) = a(n-1) + 6*n - 5 (with a(0)=0). - _Vincenzo Librandi_, Nov 20 2010

%F a(n) = 2*a(n-1) - a(n-2) + 6. - _Ant King_, Sep 01 2011

%F a(n) = A000217(n) + 5*A000217(n-1). - _Vincenzo Librandi_, Nov 20 2010

%F a(n) = (A185212(n) - 1) / 4. - _Reinhard Zumkeller_, Dec 20 2012

%F a(n) = A174709(6n). - _Philippe Deléham_, Mar 26 2013

%F a(n) = (2*n-1)^2 - (n-1)^2. - _Ivan N. Ianakiev_, Apr 10 2013

%F a(6*a(n) + 16*n + 1) = a(6*a(n) + 16*n) + a(6*n + 1). - _Vladimir Shevelev_, Jan 24 2014

%F a(0) = 0, a(n) = Sum_{k=0..n-1} A005408(A051162(n-1,k)), n >= 1. - _L. Edson Jeffery_, Jul 28 2014

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

%F From _Ilya Gutkovskiy_, Jul 29 2016: (Start)

%F Inverse binomial transform of A084857.

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

%F a(n) = n * A016777(n-1) = A053755(n) - A000290(n+1). - _Bruce J. Nicholson_, Aug 10 2017

%F Product_{n>=2} (1 - 1/a(n)) = 3/4. - _Amiram Eldar_, Jan 21 2021

%F 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

%F From _Leo Tavares_, Oct 31 2021: (Start)

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

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

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

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

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

%p A000567 := proc(n)

%p n*(3*n-2) ;

%p end proc:

%p seq(A000567(n), n=1..50) ;

%t Table[n (3 n - 2), {n, 0, 50}] (* _Harvey P. Dale_, May 06 2012 *)

%t Table[PolygonalNumber[RegularPolygon[8], n], {n, 0, 43}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)

%t PolygonalNumber[8, Range[0, 20]] (* _Eric W. Weisstein_, Sep 07 2017 *)

%t LinearRecurrence[{3, -3, 1}, {1, 8, 21}, {0, 20}] (* _Eric W. Weisstein_, Sep 07 2017 *)

%o (PARI) a(n)=n*(3*n-2) \\ _Charles R Greathouse IV_, Jun 10 2011

%o (PARI) vector(50, n, n--; n*(3*n-2)) \\ _G. C. Greubel_, Nov 15 2018

%o (GAP) List([0..50], n -> n*(3*n-2)); # _G. C. Greubel_, Nov 15 2018

%o (Haskell)

%o a000567 n = n * (3 * n - 2) -- _Reinhard Zumkeller_, Dec 20 2012

%o (Sage) [n*(3*n-2) for n in range(50)] # _G. C. Greubel_, Nov 15 2018

%o (Python) # Intended to compute the initial segment of the sequence, not isolated terms.

%o def aList():

%o x, y = 1, 1

%o yield 0

%o while True:

%o yield x

%o x, y = x + y + 6, y + 6

%o A000567 = aList()

%o print([next(A000567) for i in range(49)]) # _Peter Luschny_, Aug 04 2019

%o (Python) [n*(3*n-2) for n in range(50)] # _Gennady Eremin_, Mar 10 2022

%o (Magma) [n*(3*n-2) : n in [0..50]]; // _Wesley Ivan Hurt_, Oct 10 2021

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

%Y Cf. A005408 (the odd numbers).

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

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E Incorrect example removed by _Joerg Arndt_, Mar 11 2010