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A000384
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Hexagonal numbers: n*(2*n-1).
(Formerly M4108 N1705)
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196
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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
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OFFSET
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0,3
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COMMENTS
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Also a(n)=Sum(tan^2((k - 1/2)*pi/(2n)), k, 1, n); - Ignacio Larrosa, 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 (remartin(AT)fas.harvard.edu), Jan 07 2002
The power series expansion of the entropy function H(x) = (1+x)ln(1+x)+(1-x)ln(1-x) has 1/a_i as 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, hypotenus 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 colourings of any 2 colours chosen from n colours 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 E. Lozada, Sep 26 2012]
Partial sums give A002412. - Omar E. Pol, Jan 12 2013
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REFERENCES
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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. 2.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, 2010.
Paul Cooijmans, Odds.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 340
Milan Janjic, Two Enumerative Functions
Hyun Kwang Kim, On Regular Polytope Numbers
_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.
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).
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FORMULA
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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
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MAPLE
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A000384:=-(1+3*z)/(z-1)^3; [Simon Plouffe in his 1992 dissertation, dropping the initial zero.]
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MATHEMATICA
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Array[ #*(2*#-1)&, 20, 0] (* _Vladimir Orlovsky_, Apr 29 2008 *)
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PROG
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(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
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CROSSREFS
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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
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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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More terms from Gary W. Adamson, Dec 24 2006
Partially edited by Joerg Arndt, Mar 11 2010
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STATUS
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approved
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