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Search: id:A000567
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| A000567 |
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Octagonal numbers: n(3n-2). Also called star numbers. (Formerly M4493 N1901)
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+0 73
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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
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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
Comment from Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 02 2003: 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
First derivative at n of A045991 - Ross La Haye (rlahaye(AT)new.rr.com), Oct 23 2004
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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.
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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
Hyun Kwang Kim, On Regular Polytope Numbers
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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n*(3*n-2).
E.g.f. : exp(x)(x+3x^2) - Paul Barry (pbarry(AT)wit.ie), Jul 23 2003
G.f.: x*(1+5*x)/(1-x)^3.
a(n)=sum{k=1..n, 5n-4k} - Paul Barry (pbarry(AT)wit.ie), 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)
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MAPLE
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[ seq(n*(3*n-2), n=1..50) ];
A000567:=-(1+5*z)/(z-1)**3; [S. 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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CROSSREFS
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Cf. A001107, A051682, A014641, A014642, A014793, A014794, A001835, A016777.
Cf. A093563 ((6, 1) Pascal, column m=2). A016921 (differences).
Cf. A000217, A000566, A001106.
Adjacent sequences: A000564 A000565 A000566 this_sequence A000568 A000569 A000570
Sequence in context: A003249 A134862 A090206 this_sequence A124484 A137742 A075629
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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