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%I M4108 N1705 #452 Apr 05 2024 11:07:27
%S 0,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,
%T 780,861,946,1035,1128,1225,1326,1431,1540,1653,1770,1891,2016,2145,
%U 2278,2415,2556,2701,2850,3003,3160,3321,3486,3655,3828,4005,4186,4371,4560
%N Hexagonal numbers: a(n) = n*(2*n-1).
%C 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
%C 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
%C 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
%C Sequence also gives the 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
%C Number of divisors of 12^(n-1), i.e., A000005(A001021(n-1)). - _Henry Bottomley_, Oct 22 2001
%C 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
%C Number of standard tableaux of shape (2n-1,1,1) (n>=1). - _Emeric Deutsch_, May 30 2004
%C 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.
%C 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
%C a(n) is also a perfect number A000396 when n is an even superperfect number A061652. - _Omar E. Pol_, Sep 05 2008
%C 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
%C Let Hex(n)=hexagonal number, T(n)=triangular number, then Hex(n)=T(n)+3*T(n-1). - _Vincenzo Librandi_, Nov 10 2010
%C For n>=1, 1/a(n) = Sum_{k=0..2*n-1} ((-1)^(k+1)*binomial(2*n-1,k)*binomial(2*n-1+k,k)*H(k)/(k+1)) with H(k) harmonic number of order k.
%C 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
%C Central terms of the triangle in A051173. - _Reinhard Zumkeller_, Apr 23 2011
%C 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
%C a(n) is the number of positions of one domino in an even pyramidal board with base 2n. - _César Eliud Lozada_, Sep 26 2012
%C Partial sums give A002412. - _Omar E. Pol_, Jan 12 2013
%C 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
%C a(n+1) = A128918(2*n+1). - _Reinhard Zumkeller_, Oct 13 2013
%C 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
%C 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
%C 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
%C Binomial transform of (0, 1, 4, 0, 0, 0, ...) and second partial sum of (0, 1, 4, 4, 4, ...). - _Gary W. Adamson_, Oct 05 2015
%C a(n) also gives the dimension of the simple Lie algebras D_n, for n >= 4. - _Wolfdieter Lang_, Oct 21 2015
%C For n > 0, a(n) equals the number of compositions of n+11 into n parts avoiding parts 2, 3, 4. - _Milan Janjic_, Jan 07 2016
%C Also the number of minimum dominating sets and maximal irredundant sets in the n-cocktail party graph. - _Eric W. Weisstein_, Jun 29 and Aug 17 2017
%C As Beedassy's formula shows, this Hexagonal number sequence is the odd bisection of the Triangle number sequence. Both of these sequences are figurative number sequences. For A000384, a(n) can be found by multiplying its triangle number by its hexagonal number. For example let's use the number 153. 153 is said to be the 17th triangle number but is also said to be the 9th hexagonal number. Triangle(17) Hexagonal(9). 17*9=153. Because the Hexagonal number sequence is a subset of the Triangle number sequence, the Hexagonal number sequence will always have both a triangle number and a hexagonal number. n* (2*n-1) because (2*n-1) renders the triangle number. - _Bruce J. Nicholson_, Nov 05 2017
%C Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central valley and the largest Dyck path has a central peak, n >= 1. Thus all hexagonal numbers > 0 have middle divisors. (Cf. A237593.) - _Omar E. Pol_, Aug 28 2018
%C k^a(n-1) mod n = 1 for prime n and k=2..n-1. - _Joseph M. Shunia_, Feb 10 2019
%C Consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z: a(n+1) gives the semiperimeter of related triangles; A005408, A046092 and A001844 give the X, Y and Z values. - _Ralf Steiner_, Feb 25 2020
%C See A002939(n) = 2*a(n) for the corresponding perimeters. - _M. F. Hasler_, Mar 09 2020
%C It appears that these are the numbers k with the property that the smallest subpart in the symmetric representation of sigma(k) is 1. - _Omar E. Pol_, Aug 28 2021
%C The above conjecture is true. See A280851 for a proof. - _Hartmut F. W. Hoft_, Feb 02 2022
%C The n-th hexagonal number equals the sum of the n consecutive integers with the same parity starting at 2*n-1; for example, 1, 2+4, 3+5+7, 4+6+8+10, etc. In general, the n-th 2k-gonal number is the sum of the n consecutive integers with the same parity starting at (k-2)*n - (k-3). When k = 1 and 2, this result generates the positive integers, A000027, and the squares, A000290, respectively. - _Charlie Marion_, Mar 02 2022
%C Conjecture: For n>0, min{k such that there exist subsets A,B of {0,1,2,...,a(n)} such that |A|=|B|=k and A+B={0,1,2,...,2*a(n)}} = 2*n. - _Michael Chu_, Mar 09 2022
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78. (In the integral formula on p. 77 a left bracket is missing for the cosine argument.)
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Daniel Mondot, <a href="/A000384/b000384.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms by T. D. Noe)
%H C. K. Cook and M. R. Bacon, <a href="https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343.
%H Elena Deza and Michel Deza, <a href="https://www.fields.utoronto.ca/programs/scientific/11-12/Mtl-To-numbertheory/slides/Deza.pdf">Figurate Numbers: presentation of a book</a>, 3rd Montreal-Toronto Workshop in Number Theory, October 7-9, 2011.
%H Anicius Manlius Severinus Boethius, <a href="https://archive.org/stream/aniciimanliitor01friegoog#page/n114/mode/2up">De institutione arithmetica</a>, Book 2, section 15.
%H Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, <a href="http://www.carmamaths.org/resources/jon/walks.pdf">Random Walk Integrals</a>, The Ramanujan Journal, October 2011, 26:109. DOI: 10.1007/s11139-011-9325-y.
%H Cesar Ceballos and Viviane Pons, <a href="https://arxiv.org/abs/2309.14261">The s-weak order and s-permutahedra II: The combinatorial complex of pure intervals</a>, arXiv:2309.14261 [math.CO], 2023. See p. 41.
%H Paul Cooijmans, <a href="http://web.archive.org/web/20050302174449/http://members.chello.nl/p.cooijmans/gliaweb/tests/odds.html">Odds</a>.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>
%H Tomislav Došlić and Luka Podrug, <a href="https://arxiv.org/abs/2304.12121">Sweet division problems: from chocolate bars to honeycomb strips and back</a>, arXiv:2304.12121 [math.CO], 2023.
%H Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, <a href="https://arxiv.org/abs/2307.13749">Semi-simplicial combinatorics of cyclinders and subdivisions</a>, arXiv:2307.13749 [math.CO], 2023. See p. 32.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=340">Encyclopedia of Combinatorial Structures 340</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Pakawut Jiradilok and Elchanan Mossel, <a href="https://arxiv.org/abs/2402.11990">Gaussian Broadcast on Grids</a>, arXiv:2402.11990 [cs.IT], 2024. See p. 27.
%H Sameen Ahmed Khan, <a href="https://doi.org/10.12732/ijam.v33i2.6">Sums of the powers of reciprocals of polygonal numbers</a>, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
%H Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H Peter D. Loly and Ian D. Cameron, <a href="https://arxiv.org/abs/2008.11020">Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy</a>, arXiv:2008.11020 [math.HO], 2020.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567</a>.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H J. C. Su, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Su/su.html">On some properties of two simultaneous polygonal sequences</a>, JIS 10 (2007) 07.10.4, example 4.6.
%H Leo Tavares, <a href="/A000384/a000384.jpg">Illustration: Rectangles</a>
%H A. J. Turner and J. F. Miller, <a href="http://andrewjamesturner.co.uk/files/YDS2014.pdf">Recurrent Cartesian Genetic Programming Applied to Famous Mathematical Sequences</a>, 2014.
%H Michel Waldschmidt, <a href="http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ContinuedFractionsOujda2015.pdf">Continued fractions</a>, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>
%H Thomas Wieder, <a href="http://www.math.nthu.edu.tw/~amen/2008/070301.pdf">The number of certain k-combinations of an n-set</a>, Applied Mathematics Electronic Notes, vol. 8 (2008), pp. 45-52.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = Sum_{k=1..n} tan^2((k - 1/2)*Pi/(2n)). - _Ignacio Larrosa Cañestro_, Apr 17 2001
%F E.g.f.: exp(x)*(x+2x^2) - _Paul Barry_, Jun 09 2003
%F G.f.: x*(1+3*x)/(1-x)^3. - _Simon Plouffe_ in his 1992 dissertation, dropping the initial zero
%F a(n) = A000217(2*n-1) = A014105(-n).
%F a(n) = 4*A000217(n-1) + n. - _Lekraj Beedassy_, Jun 03 2004
%F 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
%F Row sums of triangle A131914. - _Gary W. Adamson_, Jul 27 2007
%F Row sums of n-th row, triangle A134234 starting (1, 6, 15, 28, ...). - _Gary W. Adamson_, Oct 14 2007
%F 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
%F 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; 28^2 + 29^2 + 30^2 + 31^2 = 33^2 + 34^2 + 35^2 + 4^2. - _Charlie Marion_, Nov 10 2007
%F a(n) = binomial(n+1,2) + 3*binomial(n,2).
%F 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
%F a(n) = a(n-1) + 4*n - 3 (with a(0)=0). - _Vincenzo Librandi_, Nov 20 2010
%F a(n) = A007606(A000290(n)). - _Reinhard Zumkeller_, Feb 12 2011
%F a(n) = 2*a(n-1) - a(n-2) + 4. - _Ant King_, Aug 26 2011
%F a(n+1) = A045896(2*n). - _Reinhard Zumkeller_, Dec 12 2011
%F a(2^n) = 2^(2n+1) - 2^n. - _Ivan N. Ianakiev_, Apr 13 2013
%F a(n) = binomial(2*n,2). - _Gary Detlefs_, Jul 28 2013
%F a(4*a(n)+7*n+1) = a(4*a(n)+7*n) + a(4*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F Sum_{n>=1} 1/a(n) = 2*log(2) = 1.38629436111989...= A016627. - _Vaclav Kotesovec_, Apr 27 2016
%F Sum_{n>=1} (-1)^n/a(n) = log(2) - Pi/2. - _Vaclav Kotesovec_, Apr 20 2018
%F a(n+1) = trinomial(2*n+1, 2) = trinomial(2*n+1, 4*n), for n >= 0, with the trinomial irregular triangle A027907. a(n+1) = (n+1)*(2*n+1) = (1/Pi)*Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n+1)*R(4*n-2, x) with the R polynomial coefficients given in A127672. [Comtet, p. 77, the integral formula for q=3, n -> 2*n+1, k = 2, rewritten with x = 2*cos(phi)]. - _Wolfdieter Lang_, Apr 19 2018
%F Sum_{n>=1} 1/(a(n))^2 = 2*Pi^2/3-8*log(2) = 1.0345588... = 10*A182448 - A257872. - _R. J. Mathar_, Sep 12 2019
%F a(n) = (A005408(n-1) + A046092(n-1) + A001844(n-1))/2. - _Ralf Steiner_, Feb 27 2020
%F Product_{n>=2} (1 - 1/a(n)) = 2/3. - _Amiram Eldar_, Jan 21 2021
%F a(n) = floor(Sum_{k=(n-1)^2..n^2} sqrt(k)), for n >= 1. - _Amrit Awasthi_, Jun 13 2021
%F a(n+1) = A084265(2*n), n>=0. - _Hartmut F. W. Hoft_, Feb 02 2022
%F a(n) = A000290(n) + A002378(n-1). - _Charles Kusniec_, Sep 11 2022
%p A000384:=n->n*(2*n-1); seq(A000384(k), k=0..100); # _Wesley Ivan Hurt_, Sep 27 2013
%t Table[n*(2 n - 1), {n, 0, 100}] (* _Wesley Ivan Hurt_, Sep 27 2013 *)
%t LinearRecurrence[{3, -3, 1}, {0, 1, 6}, 50] (* _Harvey P. Dale_, Sep 10 2015 *)
%t Join[{0}, Accumulate[Range[1, 312, 4]]] (* _Harvey P. Dale_, Mar 26 2016 *)
%t (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[6], n], {n, 0, 48}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t PolygonalNumber[6, Range[0, 20]] (* _Eric W. Weisstein_, Aug 17 2017 *)
%t CoefficientList[Series[x*(1 + 3*x)/(1 - x)^3 , {x, 0, 100}], x] (* _Stefano Spezia_, Sep 02 2018 *)
%o (PARI) a(n)=n*(2*n-1)
%o (PARI) a(n) = binomial(2*n,2) \\ _Altug Alkan_, Oct 06 2015
%o (Haskell)
%o a000384 n = n * (2 * n - 1)
%o a000384_list = scanl (+) 0 a016813_list
%o -- _Reinhard Zumkeller_, Dec 16 2012
%o (Python) # Intended to compute the initial segment of the sequence, not isolated terms.
%o def aList():
%o x, y = 1, 1
%o yield 0
%o while True:
%o yield x
%o x, y = x + y + 4, y + 4
%o A000384 = aList()
%o print([next(A000384) for i in range(49)]) # _Peter Luschny_, Aug 04 2019
%Y Cf. A000217, A014105, A127672, A027907, A005408, A046092, A001844.
%Y a(n)= A093561(n+1, 2), (4, 1)-Pascal column.
%Y a(n) = A100345(n, n-1) for n>0.
%Y Cf. A002939 (twice a(n): sums of Pythagorean triples (X, Y, Z=Y+1).
%Y Cf. A280851.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Partially edited by _Joerg Arndt_, Mar 11 2010
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