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A000384 Hexagonal numbers: n*(2*n-1).
(Formerly M4108 N1705)
219
0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4186, 4371, 4560 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also a(n)=sum(tan^2((k - 1/2)*Pi/(2n)), k, 1, n). - Ignacio Larrosa Cañestro, Apr 17 2001

Number of edges in the join of two complete graphs, each of order n, K_n * K_n. - Roberto E. Martinez II, Jan 07 2002

The power series expansion of the entropy function H(x) = (1+x)log(1+x)+(1-x)log(1-x) has 1/a_i as the coefficient of x^(2i) (the odd terms being zero). - Tommaso Toffoli (tt(AT)bu.edu), May 06 2002

Partial sums of A016813 (4n+1). Also with offset = 0, a(n) = (2n+1)(n+1) = A005408 * A000027 = 2n^2 + 3n + 1, i.e. a(0) = 1. - Jeremy Gardiner, Sep 29 2002

Sequence also refers to greatest semiperimeter of primitive Pythagorean triangles having inradius n-1. Such a triangle has consecutive longer sides, with short leg 2n-1, hypotenuse a(n)-(n-1)=A001844(n), and area (n-1)*a(n)=6*A000330(n-1). - Lekraj Beedassy, Apr 23 2003

Number of divisors of 12^(n-1), i.e., A000005(A001021(n-1)). - Henry Bottomley, Oct 22 2001

More generally, if p1 and p2 are two arbitrarily chosen distinct primes then a(n) is the number of divisors of (p1^2*p2)^(n-1) or equivalently of any member of A054753^(n-1). - Ant King, Aug 29 2011

Number of standard tableaux of shape (2n-1,1,1) (n>=1). - Emeric Deutsch, May 30 2004

It is well known that for n>0, A014105(n) [0,3,10,21,...] is the first of 2n+1 consecutive integers such that the sum of the squares of the first n+1 such integers is equal to the sum of the squares of the last n; e.g., 10^2+11^2+12^2=13^2+14^2.

Less well known is that for n>1, a(n) [0,1,6,15,28... ] is the first of 2n consecutive integers such that sum of the squares of the first n such integers is equal to the sum of the squares of the last n-1 plus n^2; e.g. 15^2+16^2+17^2 = 19^2+20^2+3^2. - Charlie Marion, Dec 16 2006

a(n) is also a perfect number A000396 when n is an even superperfect number A061652. - Omar E. Pol, Sep 05 2008

Sequence found by reading the line from 0, in the direction 0, 6,... and the line from 1, in the direction 1, 15,..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Jan 09 2009

Let Hex(n)=hexagonal number, T(n)=triangular number, then Hex(n)=T(n)+3*T(n-1). [Vincenzo Librandi, Nov 10 2010]

For n>=1, 1/a(n)=sum((-1)^(k+1)*binomial(2*n-1,k)*binomial(2*n-1+k,k)*H(k)/(k+1),k=0..2*n-1)) with H(k) harmonic number of order k.

The number of possible distinct colorings of any 2 colors chosen from n colors of a square divided into quadrants. - Paul Cleary, Dec 21 2010

Central terms of the triangle in A051173. - Reinhard Zumkeller, Apr 23 2011

a(n+1) = A045896(2*n). - Reinhard Zumkeller, Dec 12 2011

For n>0, a(n-1) is the number of triples (w,x,y) with all terms in {0,...,n} and max(|w-x|,|x-y|) = |w-y|.  [Clark Kimberling, Jun 12 2012]

a(n) is the number of positions of one domino in an even pyramidal board with base 2n. - César Eliud Lozada, Sep 26 2012

Partial sums give A002412. - Omar E. Pol, Jan 12 2013

Let a triangle have T(0,0) = 0 and T(r,c) = |r^2 - c^2|. The sum of the differences of the terms in row(n) and row(n-1) is a(n). - J. M. Bergot, Jun 17 2013

a(n+1) = A128918(2*n+1). - Reinhard Zumkeller, Oct 13 2013

With T_(i+1,i)=a(i+1) and all other elements of the lower triangular matrix T zero, T is the infinitesimal generator for A176230, analogous to A132440 for the Pascal matrix. - Tom Copeland, Dec 11 2013

a(n) is the number of length 2n binary sequences that have exactly two 1's. a(2) = 6 because we have: {0,0,1,1}, {0,1,0,1}, {0,1,1,0}, {1,0,0,1}, {1,0,1,0}, {1,1,0,0}.  The ordinary generating function with interpolated zeros is: (x^2 + 3*x^4)/(1-x^2)^3. - Geoffrey Critzer, Jan 02 2014

For n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n. More generally, for m > 0 and n > 0, the largest integer k such that k^(2*m) + n^(2*m) is a multiple of k + n is given by k = 2*n^(2*m) - n. - Derek Orr, Sep 04 2014

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

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. 2.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

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

Anicius Manlius Severinus Boethius, De institutione arithmetica, Book 2, section 15.

Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.

Paul Cooijmans, Odds.

Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 340

Milan Janjic, Two Enumerative Functions

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567

Eric Weisstein's World of Mathematics, Hexagonal Number

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

Index entries for two-way infinite sequences

Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

E.g.f.: exp(x)*(x+2x^2) - Paul Barry, Jun 09 2003

G.f.: x*(1+3*x)/(1-x)^3.

a(n) = A000217(2*n-1) = A014105(-n).

a(n) = 4*A000217(n-1) + n. - Lekraj Beedassy, Jun 03 2004

a(n) = right term of M^n * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,4,1]. Example: a(5) = 45 since M^5 *[1,0,0] = [1,5,45]. - Gary W. Adamson, Dec 24 2006

Row sums of triangle A131914. - Gary W. Adamson, Jul 27 2007

Row sums of n-th row, triangle A134234 starting (1, 6, 15, 28,...). - Gary W. Adamson, Oct 14 2007

Starting with offset 1, = binomial transform of [1, 5, 4, 0, 0, 0,...]. Also, A004736 * [1, 4, 4, 4,...]. - Gary W. Adamson, Oct 25 2007

a(n)^2+(a(n)+1)^2+...+(a(n)+n-1)^2=(a(n)+n+1)^2+...(a(n)+2n-1)^2+n^2; e.g., 6^2+7^2=9^2+2^2; 28^+29^2+30^2+31^2=33^2+34^2+35^2+4^2. - Charlie Marion, Nov 10 2007

a(n) = C(n+1,2) + 3*C(n,2).

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=6. - Jaume Oliver Lafont, Dec 02 2008

a(n) = a(n-1)+4*n-3 (with a(0)=0). - Vincenzo Librandi, Nov 20 2010

a(n) = A007606(A000290(n)). - Reinhard Zumkeller, Feb 12 2011

a(n) = 2*a(n-1)-a(n-2)+4. - Ant King, Aug 26 2011

a(2^n) = 2^(2n+1) - 2^n. - Ivan N. Ianakiev, Apr 13 2013

a(n) = binomial(2*n,2). - Gary Detlefs, Jul 28 2013

a(4*a(n)+7*n+1) =  a(4*a(n)+7*n) + a(4*n+1). - Vladimir Shevelev, Jan 24 2014

MAPLE

A000384:=-(1+3*z)/(z-1)^3; # [Simon Plouffe in his 1992 dissertation, dropping the initial zero.]

A000384:=n->n*(2*n-1); seq(A000384(k), k=0..100); # Wesley Ivan Hurt, Sep 27 2013

MATHEMATICA

Table[n*(2n-1), {n, 0, 100}] (* Wesley Ivan Hurt, Sep 27 2013 *)

PROG

(PARI) a(n)=n*(2*n-1)

(Haskell)

a000384 n = n * (2 * n - 1)

a000384_list = scanl (+) 0 a016813_list

-- Reinhard Zumkeller, Dec 16 2012

CROSSREFS

Cf. A014105.

a(n)= A093561(n+1, 2), (4, 1)-Pascal column.

a(n)=A100345(n, n-1) for n>0. Cf. n-gonal numbers: A000217, A000290, A000326, this sequence, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.

Sequence in context: A094142 A081873 A096892 * A164000 A212087 A134978

Adjacent sequences:  A000381 A000382 A000383 * A000385 A000386 A000387

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Gary W. Adamson, Dec 24 2006

Partially edited by Joerg Arndt, Mar 11 2010

Broken link to Hyun Kwang Kim's paper fixed by Felix Fröhlich, Jun 16 2014

STATUS

approved

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Last modified September 22 13:57 EDT 2014. Contains 247065 sequences.