%I A003215 M4362
%S A003215 1,7,19,37,61,91,127,169,217,271,331,397,469,547,631,721,817,919,1027,
%T A003215 1141,1261,1387,1519,1657,1801,1951,2107,2269,2437,2611,2791,2977,3169,
%U A003215 3367,3571,3781,3997,4219,4447,4681,4921,5167,5419,5677,5941,6211,6487
%N A003215 Hex (or centered hexagonal) numbers: 3n(n+1)+1 (crystal ball sequence 
               for hexagonal lattice).
%C A003215 The hexagonal lattice is the familiar 2-dimensional lattice in which 
               each point has 6 neighbors. This is sometimes called the triangular 
               lattice.
%C A003215 Sixth spoke of hexagonal spiral (cf. A056105-A056109).
%C A003215 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
%C A003215 Also the number of partitions of 6n into at most 3 parts. - R. K. Guy, 
               Oct 20, 2003
%C A003215 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
%C A003215 Number of dots in a centered hexagonal figure with n+1 dots on each side.
%C A003215 Values of second Bessel polynomial y_2(n) (see A001498).
%C A003215 First differences of the cubes. - Allan Turton (a_turton(AT)origo.com.au), 
               May 15 2006
%C A003215 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
%C A003215 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
%C A003215 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
%C A003215 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
%C A003215 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]
%D A003215 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003215 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.
%D A003215 B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 
               7 (1967), 23-31 (see p. 30).
%D A003215 M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, 
               NY, 1988, p. 18.
%D A003215 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), 
               no. 8, 697-712.
%D A003215 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)
%D A003215 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral 
               clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A003215 T. D. Noe, <a href="b003215.txt">Table of n, a(n) for n=0..1000</a>
%H A003215 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A003215 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 A003215 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 A003215 H. Bottomley, <a href="a3215.gif">Illustration of initial terms</a>
%H A003215 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination 
               Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http:/
               /www.research.att.com/~njas/doc/ldl7.txt">Abstract</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.pdf">pdf</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.ps">ps</a>).
%H A003215 G. Nebe and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               lattices/A2.html">Home page for hexagonal (or triangular) lattice 
               A2</a>
%H A003215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexNumber.html">Link to a section of The World of Mathematics.</a>
%H A003215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NexusNumber.html">Nexus Number</a>
%H A003215 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OuterNapoleonTriangle.html">Outer Napoleon Triangle</a>.
%H A003215 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related 
               to centered polygonal numbers</a>
%H A003215 <a href="Sindx_Cor.html#crystal_ball">Index entries for crystal ball 
               sequences</a>
%H A003215 <a href="Sindx_Aa.html#A2">Index entries for sequences related to A2 
               = hexagonal = triangular lattice</a>
%F A003215 a(n)=(n+1)^3-n^3. G.f.: (1+4*x+x^2)/(1-x)^3.
%F A003215 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
%F A003215 n-th partial arithmetic mean is n^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               May 27 2003
%F A003215 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
%F A003215 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
%F A003215 First differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), 
               Dec 15 2004
%F A003215 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
%F A003215 Row sums of triangle A130298. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 07 2007
%F A003215 a(n) = A132111(n+1,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 10 2007
%F A003215 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
%F A003215 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
%F A003215 a(n)=6*n+a(n-1)-6 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 09 2009]
%e A003215 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]
%p A003215 A003215:=-(1+4*z+z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%t A003215 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]
%t A003215 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]
%o A003215 (PARI) a(n)=3*n*(n+1)+1
%o A003215 (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
%Y A003215 A003215(n)=6*A000217(n)+1. Cf. A028896, A003154, A005891, A063496.
%Y A003215 Column T(n, 3) of A080853
%Y A003215 Cf. A000578.
%Y A003215 Cf. A130298.
%Y A003215 Cf. A001263.
%Y A003215 Sequence in context: A136057 A023224 A113743 this_sequence A133323 A002407 
               A098484
%Y A003215 Adjacent sequences: A003212 A003213 A003214 this_sequence A003216 A003217 
               A003218
%K A003215 nonn,easy,nice,new
%O A003215 0,2
%A A003215 N. J. A. Sloane (njas(AT)research.att.com).
%E A003215 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Mar 11 2009