0,2

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

Sixth spoke of hexagonal spiral (cf. A056105-A056109).

Number of ordered triples (a,b,c), -n<= a,b,c <=n, such that a+b+c=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 14 2003

Also the number of partitions of 6n into at most 3 parts. - R. K. Guy, Oct 20, 2003

Also, a(n) is the number of partitions of 6(n+1) into exactly 3 distinct parts. - William J. Keith (keith(AT)math.psu.edu), Jul 01 2004

Number of dots in a centered hexagonal figure with n+1 dots on each side.

Values of second Bessel polynomial y_2(n) (see A001498).

First differences of the cubes. - Allan Turton (a_turton(AT)origo.com.au), May 15 2006

Final digits of Hex numbers Mod[Hex[n], 10] are periodic with palindromic period of length 5 {1, 7, 9, 7, 1}. Last two digits of Hex numbers Mod[Hex[n], 100] are periodic with palindromic period of length 100. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 11 2006

All divisors of a(n) are congruent to 1, modulo 6. Proof: If p is an odd prime different from 3 then 3n^2 + 3n + 1 = 0 (mod p) implies 9(2n + 1)^2 = -3 (mod p), whence p = 1 (mod 6). - Nick Hobson Nov 13 2006

For n>=1, a(n) = side of Outer Naploleon Triangle whose reference triangle is a right triangle with legs (3a(n))^(1/2) and 3n(a(n))^(1/2). - Tom Schicker (tschicke(AT)email.smith.edu), Apr 25 2007

Number of triples (a,b,) where 0<=(a,b)<=n and c=n (at least once the term n). E.g. for n = 1 : (0,0,1),0,1,0),(1,0,0),(0,1,1),(1,0,1),(1,1,0),(1,1,1), then c(1)=7 - Philippe Lalloouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007

Equals the triangular numbers convolved with [1, 4, 1, 0, 0, 0,...]. [From Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), May 29 2009]

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

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.

B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31 (see p. 30).

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 18.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

G. S. Kazandzidis, On a Conjecture of Moessner and a General Problem, Bull. Soc. Math. Grece, Nouvelle Serie - vol. 2, fasc. 1-2, pp. 23-30.(1961)

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

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

Index entries for sequences related to linear recurrences with constant coefficients

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.

H. Bottomley, Illustration of initial terms

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Nexus Number

Eric Weisstein's World of Mathematics, Outer Napoleon Triangle.

Index entries for sequences related to centered polygonal numbers

Index entries for crystal ball sequences

Index entries for sequences related to A2 = hexagonal = triangular lattice

a(n)=(n+1)^3-n^3. G.f.: (1+4*x+x^2)/(1-x)^3.

a(n) = a(n-1)+6n = 2a(n-1)-a(n-2)+6 = 3a(n-1)-3a(n-2)+a(n-3) = A056105(n)+5n = A056106(n)+4n = A056107(n)+3n = A056108(n)+2n = A056108(n)+n

n-th partial arithmetic mean is n^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 27 2003

a(n) = 1 + (sum(6*n)). E.g. a(2)=19 because 1+ 6*0 + 6*1 + 6*2 =19. - Xavier Acloque, Oct 06 2003

The sum of the first n hexagonal numbers is n^3. That is, sum[ 3n(n-1)+1 ] = n^3. - Edward Weed (eweed(AT)gdrs.com), Oct 23 2003

First differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 1 0 / 3 3 1]. M^n * [1 1 1] = [1 2n+1 a(n)]. E.g. a(4) = 61, right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 9 61] = [1 2n+1 a(4)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 22 2004

Row sums of triangle A130298. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 07 2007

a(n) = A132111(n+1,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 10 2007

c(n)=3*n^2+3*n+1. Proof : 1) if n occurs once, it may be in 3 positions; for the two other ones,n terms are independently possible, then we have 3*n^2 different triples 2) If the term n occurs twice, the third one may be placed in 3 positions and have n possible values, then we have 3*n more different triples 3) The term n may occurs 3 times in one way only That gives the formula. - Philippe Lalloouet (philip.lallouet(AT)wanadoo.fr), Aug 20 2007

Binomial transform of [1, 6, 6, 0, 0, 0,...]; Narayana transform (A001263) of [1, 6, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2007

a(n)=6*n+a(n-1)-6 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]

For n=2, a(2)=6*2+1-6=7; n=3, a(3)=6*3+7-6=19; n=4, a(4)=6*4+19-6=37 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]

A003215:=-(1+4*z+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

s=1; lst={}; Do[s+=2*n; AppendTo[lst, s], {n, 0, 6!, 3}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 10 2008]

a[n_]:=(n+1)^3-n^3; lst={}; Do[AppendTo[lst, a[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 20 2008]

(PARI) a(n)=3*n*(n+1)+1

(Sage) def sd(seq): return [seq[i+1] - seq[i] for i in range(len(seq)-1)] sd([i^3 for i in range(0, 19)]) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 27 2007

A003215(n)=6*A000217(n)+1. Cf. A028896, A003154, A005891, A063496.

Column T(n, 3) of A080853

Cf. A000578.

Cf. A130298.

Cf. A001263.

Sequence in context: A136057 A023224 A113743 this_sequence A133323 A002407 A098484

Adjacent sequences: A003212 A003213 A003214 this_sequence A003216 A003217 A003218

nonn,easy,nice,new

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

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009