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A000567
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Octagonal numbers: n*(3*n-2). Also called star numbers.
(Formerly M4493 N1901)
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138
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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
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OFFSET
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0,3
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COMMENTS
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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 (amarnath_murthy(AT)yahoo.com), 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. - 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
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
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.
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.
L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.
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.
Viktor Levandovskyy, Christoph Koutschan and Oleksandr Motsak, On Two-generated Non-commutative Algebras Subject to the Affine Relation, arXiv:1108.1108, 2011.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 342
M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550, 2013
Hyun Kwang Kim, On Regular Polytope Numbers
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 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).
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FORMULA
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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, 01 Sep 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
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MAPLE
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[ 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 (zerinvarylajos(AT)yahoo.com), Feb 18 2008
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MATHEMATICA
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Table[n(3n-2), {n, 0, 50}] (* Harvey P. Dale, May 06 2012 *)
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PROG
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(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
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CROSSREFS
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Cf. A014641, A014642, A014793, A014794, A001835, A016777.
Cf. A093563 ((6, 1) Pascal, column m=2). A016921 (differences).
Cf. A045944.
Cf. n-gonal numbers: A000217, A000290, A000326, A000566, A001106, A001107, A051682, A051624, A051865-A051876.
Sequence in context: A179681 A224039 A225287 * A124484 A137742 A190456
Adjacent sequences: A000564 A000565 A000566 * A000568 A000569 A000570
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Removed incorrect example Joerg Arndt, Mar 11 2010
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STATUS
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approved
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