0,3

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

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 - J. Lowell (jhbubby(AT)mindspring.com), Aug 30 2008

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]

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]

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.

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

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.

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)

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

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]

[ 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

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]

Cf. A001107, A051682, A014641, A014642, A014793, A014794, A001835, A016777.

Cf. A093563 ((6, 1) Pascal, column m=2). A016921 (differences).

Cf. A000217, A000566, A001106.

Cf. A045944.

Adjacent sequences: A000564 A000565 A000566 this_sequence A000568 A000569 A000570

Sequence in context: A090206 A139590 A154894 this_sequence A124484 A137742 A152117

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).