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A003215
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Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).
(Formerly M4362)
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123
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1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487
(list;
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OFFSET
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0,2
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COMMENTS
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The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Crystal ball sequence for A_2 lattice. - Michael Somos, Jun 03 2012
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, Jun 14 2003
Also the number of partitions of 6n into at most 3 parts, A001399(6n). - 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, 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 cubes (A000578). - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
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, 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 Napoleon 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,...]. [Gary W. Adamson and Alexander R. Povolotsky, May 29 2009]
Comment from Terry Stickels, Dec 07 2009: (Start) The sequence : 1, 7, 19, 37, 61, 91, is also the maximum number of viewable cubes from any one static point while viewing cube stack of identical cubes of varying magnitude.
For example, viewing a 2 x 2 x 2 stack will yield 7 maximum viewable cubes.
If the stack is 3 x 3 x 3 , the maximum number of viewable cubes from any one static position is 19, and so on.
The number of cubes in the stack must always be the same number for width, length, height ( at true regular cubic stack) and the maximum number of visible cubes can always be found by talking any cubic number and subtracting the number of the cube that is one less.
Examples: 125 - 64 = 61, 64 - 27 = 37, 27 - 8 = 19. (End)
Contribution from Gary Detlefs, Dec 06 2009: (Start)
a(n) = (n-1)*A000166(n) + (n-2)*A000166(n-1)
ie..31= 3*9+2*4, 203=4*44+3*9...A function of !n (End)
The sequence of digital roots of the a(n) is period 3: repeat [1,7,1]. - Ant King, Jun 17 2012
The average of the first n (n>0) centered hexagonal numbers is the n-th square. - Philippe Deléham, Feb 04 2013
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REFERENCES
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Alexanderson, G. L.; Wetzel, John E. Dissections of a tetrahedron. J. Combinatorial Theory Ser. B 11 (1971), 58--66. MR0303412 (46 #2549). See p. 58.
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)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
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
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Hex Number
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
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = (n+1)^3 - n^3 = a(-1-n).
G.f.: (1 + 4*x + x^2) / (1 - x)^3.
a(n) = a(n-1)+6*n = 2a(n-1)-a(n-2)+6 = 3*a(n-1)-3*a(n-2)+a(n-3) = A056105(n)+5n = A056106(n)+4*n = A056107(n)+3*n = A056108(n)+2*n = 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_{j=0..n} (6*j). 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(n>=1, 3*n*(n-1)+1 ) = n^3. - Edward Weed (eweed(AT)gdrs.com), Oct 23 2003
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, Dec 22 2004
Row sums of triangle A130298. - Gary W. Adamson, Jun 07 2007
a(n) = A132111(n+1,n) for n>0. - Reinhard Zumkeller, Aug 10 2007
a(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, Dec 29 2007
a(n) = (n-1)floor(n!*e^(-1)+1) + (n-2)*floor((n-1)!*e^(-1)+1) (with offset 0) [From Gary Detlefs, Dec 06 2009]
a(n) = A028896(n) + 1. - Omar E. Pol, Oct 03 2011
a(n)= integral( (sin((n+1/2)x)/sin(x/2))^3, x=0..Pi)/Pi. - Aktar Yalcin, Dec 03 2011
Sum(n>=1,1/a(n)) = pi/sqrt(3)*tanh(pi/(2*sqrt(3))) = 1.305284153013581... - Ant King, Jun 17 2012
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EXAMPLE
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For n=1, a(1)=6*1+1=7; a(2)=6*2+7=19; a(3)=6*3+19=37
1 + 7*x + 19*x^2 + 37*x^3 + 61*x^4 + 91*x^5 + 127*x^6 + 169*x^7 + 217*x^8 + ...
Contribution from Omar E. Pol, Aug 21 2011: (Start)
Illustration of initial terms:
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. o o o o
. o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o o o o o o o o o o o
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. o o o o o o o o
. o o o o
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. 1 7 19 37
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(End)
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MAPLE
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A003215:=-(1+4*z+z**2)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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s=1; lst={}; Do[s+=2*n; AppendTo[lst, s], {n, 0, 6!, 3}]; lst (* Vladimir Orlovsky, Nov 10 2008 *)
a[n_]:=(n+1)^3-n^3; lst={}; Do[AppendTo[lst, a[n]], {n, 0, 5!}]; lst (* Vladimir Orlovsky, Dec 20 2008 *)
FoldList[#1 + #2 &, 1, 6 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
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PROG
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(PARI) {a(n) = 3*n*(n+1) + 1}
(Haskell)
a003215 n = 3 * n * (n + 1) + 1 -- Reinhard Zumkeller, Oct 22 2011
(Maxima) makelist(3*n*(n+1)+1, n, 0, 30); /* Martin Ettl, Nov 12 2012 */
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CROSSREFS
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A003215(n)=6*A000217(n)+1. Cf. A002407 (primes), A028896, A003154, A005891, A063496.
Column T(n, 3) of A080853
Cf. A000578, A001263, A048766, A220083 (for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides).
Sequence in context: A177092 A023224 A113743 * A133323 A002407 A098484
Adjacent sequences: A003212 A003213 A003214 * A003216 A003217 A003218
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Removed attribute "conjectured" from Simon Plouffe g.f. - R. J. Mathar, Mar 11 2009
Partially edited by Joerg Arndt, Mar 11 2010
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STATUS
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approved
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