%I A000567 M4493 N1901
%S A000567 0,1,8,21,40,65,96,133,176,225,280,341,408,481,560,645,736,833,
%T A000567 936,1045,1160,1281,1408,1541,1680,1825,1976,2133,2296,2465,2640,
%U A000567 2821,3008,3201,3400,3605,3816,4033,4256,4485,4720,4961,5208,5461
%N A000567 Octagonal numbers: n(3n-2). Also called star numbers.
%C A000567 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:
%C A000567 ......16..15..14
%C A000567 ....17..5...4...13
%C A000567 ..18..6...0...3...12
%C A000567 19..7...1...2...11..26
%C A000567 ..20..8...9...10..25
%C A000567 ....21..22..23..24
%C A000567 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
%C A000567 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:
%C A000567 x x
%C A000567 x x x
%C A000567 x x x x
%C A000567 x x x x x
%C A000567 x x x x x x
%C A000567 x x x x x x
%C A000567 First derivative at n of A045991 - Ross La Haye (rlahaye(AT)new.rr.com),
Oct 23 2004
%C A000567 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 A000567 Number of divisors of 24^n - J. Lowell (jhbubby(AT)mindspring.com), Aug
30 2008
%C A000567 a(n+2)=A005563(2), A061037(3), A061039(4), A061041(5), A061043(6), A061045(7),
A061047(8), A061049(9), .. . From respective Lyman, Balmer, Paschen,
Brackett, Pfund, Humphreys, Hansen-Strong, .. spectra of hydrogen.
[From Paul Curtz (bpcrtz(AT)free.fr), Oct 08 2008]
%C A000567 Also, let Oct(n)=octagonal numbers, T(n)=triangular numbers, then Oct(n)=T(n)+5*T(n-1)
[From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 28 2009]
%C A000567 a(n) = A000578(n) - A007531(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Sep 18 2009]
%D A000567 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000567 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000567 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964,
p. 189.
%D A000567 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 A000567 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.
%D A000567 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer
Press, NY, 1950, p. 36.
%H A000567 T. D. Noe, <a href="b000567.txt">Table of n, a(n) for n=0..1000</a>
%H A000567 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A000567 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=342">
Encyclopedia of Combinatorial Structures 342</a>
%H A000567 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
On Regular Polytope Numbers</a>
%H A000567 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A000567 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000567 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
OctagonalNumber.html">Link to a section of The World of Mathematics.</
a>
%F A000567 n*(3*n-2).
%F A000567 E.g.f. : exp(x)(x+3x^2) - Paul Barry (pbarry(AT)wit.ie), Jul 23 2003
%F A000567 G.f.: x*(1+5*x)/(1-x)^3.
%F A000567 a(n)=sum{k=1..n, 5n-4k} - Paul Barry (pbarry(AT)wit.ie), Sep 06 2005
%F A000567 a(n)=n+6*A000217(n-1) - Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct
14 2005
%F A000567 a(n) = C(n+1,2) + 5 C(n,2)
%F A000567 Starting (1, 8, 21, 40, 65,...) = binomial transform of [1, 7, 6, 0,
0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 30 2008
%F A000567 a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=8 [From Jaume Oliver
Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
%F A000567 a(n)=6*n+a(n-1)-11 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 08 2009]
%e A000567 For n=2, a(2)=6*2+0-11=1; n=3, a(3)=6*3+1-11=8; n=4, a(4)=6*4+8-11=21
[From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A000567 [ seq(n*(3*n-2), n=1..50) ];
%p A000567 A000567:=-(1+5*z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A000567 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
%t A000567 s=0;lst={s};Do[s+=n+++1;AppendTo[lst, s], {n, 0, 6!, 6}];lst [From Vladimir
Orlovsky (4vladimir(AT)gmail.com), Nov 15 2008]
%Y A000567 Cf. A001107, A051682, A014641, A014642, A014793, A014794, A001835, A016777.
%Y A000567 Cf. A093563 ((6, 1) Pascal, column m=2). A016921 (differences).
%Y A000567 Cf. A000217, A000566, A001106.
%Y A000567 Cf. A045944.
%Y A000567 Sequence in context: A090206 A139590 A154894 this_sequence A124484 A137742
A152117
%Y A000567 Adjacent sequences: A000564 A000565 A000566 this_sequence A000568 A000569
A000570
%K A000567 nonn,easy,nice,new
%O A000567 0,3
%A A000567 N. J. A. Sloane (njas(AT)research.att.com).
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