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A000384 Hexagonal numbers: a(n) = n*(2*n-1).

%I A000384 M4108 N1705 #491 Aug 12 2025 06:49:11
%S A000384 0,1,6,15,28,45,66,91,120,153,190,231,276,325,378,435,496,561,630,703,
%T A000384 780,861,946,1035,1128,1225,1326,1431,1540,1653,1770,1891,2016,2145,
%U A000384 2278,2415,2556,2701,2850,3003,3160,3321,3486,3655,3828,4005,4186,4371,4560
%N A000384 Hexagonal numbers: a(n) = n*(2*n-1).
%C A000384 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 A000384 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 A000384 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 A000384 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 A000384 Number of divisors of 12^(n-1), i.e., A000005(A001021(n-1)). - _Henry Bottomley_, Oct 22 2001
%C A000384 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 A000384 Number of standard tableaux of shape (2n-1,1,1) (n>=1). - _Emeric Deutsch_, May 30 2004
%C A000384 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 A000384 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 A000384 a(n) is also a perfect number A000396 when n is an even superperfect number A061652. - _Omar E. Pol_, Sep 05 2008
%C A000384 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 A000384 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 A000384 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 A000384 Central terms of the triangle in A051173. - _Reinhard Zumkeller_, Apr 23 2011
%C A000384 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 A000384 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 A000384 Partial sums give A002412. - _Omar E. Pol_, Jan 12 2013
%C A000384 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 A000384 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 A000384 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 A000384 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 A000384 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 A000384 a(n) also gives the dimension of the simple Lie algebras D_n, for n >= 4. - _Wolfdieter Lang_, Oct 21 2015
%C A000384 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 A000384 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 A000384 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 A000384 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 A000384 k^a(n-1) mod n = 1 for prime n and k=2..n-1. - _Joseph M. Shunia_, Feb 10 2019
%C A000384 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 A000384 See A002939(n) = 2*a(n) for the corresponding perimeters. - _M. F. Hasler_, Mar 09 2020
%C A000384 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 A000384 The above conjecture is true. See A280851 for a proof.  - _Hartmut F. W. Hoft_, Feb 02 2022
%C A000384 The n-th hexagonal number equals the sum of the n consecutive integers with the same parity starting at n; 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 A000384 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 A000384 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%D A000384 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 A000384 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
%D A000384 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
%D A000384 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 A000384 Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 53-54, 129-130, 132.
%D A000384 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000384 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000384 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 21.
%D A000384 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 122-123.
%H A000384 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 A000384 C. K. Cook and M. R. Bacon, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-4/CookBacon4292014.pdf">Some polygonal number summation formulas</a>, Fib. Q., 52 (2014), 336-343.
%H A000384 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 A000384 Anicius Manlius Severinus Boethius, <a href="https://archive.org/stream/aniciimanliitor01friegoog#page/n114/mode/2up">De institutione arithmetica</a>, Book 2, section 15.
%H A000384 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 A000384 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 A000384 Paul Cooijmans, <a href="http://web.archive.org/web/20050302174449/http://members.chello.nl/p.cooijmans/gliaweb/tests/odds.html">Odds</a>.
%H A000384 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 A000384 Olivier Danvy, <a href="https://arxiv.org/abs/2412.03127">Summa Summarum: Moessner's Theorem without Dynamic Programming</a>, arXiv:2412.03127 [cs.DM], 2024. See p. 33.
%H A000384 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 A000384 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 A000384 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=340">Encyclopedia of Combinatorial Structures 340</a>.
%H A000384 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A000384 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 A000384 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 A000384 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 A000384 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 A000384 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 A000384 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 A000384 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A000384 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 A000384 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 A000384 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 A000384 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 A000384 Leo Tavares, <a href="/A000384/a000384.jpg">Illustration: Rectangles</a>.
%H A000384 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 A000384 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 A000384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H A000384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>.
%H A000384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>.
%H A000384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalIrredundantSet.html">Maximal Irredundant Set</a>.
%H A000384 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 A000384 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.
%H A000384 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H A000384 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A000384 a(n) = Sum_{k=1..n} tan^2((k - 1/2)*Pi/(2n)). - _Ignacio Larrosa Cañestro_, Apr 17 2001
%F A000384 E.g.f.: exp(x)*(x+2x^2). - _Paul Barry_, Jun 09 2003
%F A000384 G.f.: x*(1+3*x)/(1-x)^3. - _Simon Plouffe_ in his 1992 dissertation, dropping the initial zero
%F A000384 a(n) = A000217(2*n-1) = A014105(-n).
%F A000384 a(n) = 4*A000217(n-1) + n. - _Lekraj Beedassy_, Jun 03 2004
%F A000384 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 A000384 Row sums of triangle A131914. - _Gary W. Adamson_, Jul 27 2007
%F A000384 Row sums of n-th row, triangle A134234 starting (1, 6, 15, 28, ...). - _Gary W. Adamson_, Oct 14 2007
%F A000384 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 A000384 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 A000384 a(n) = binomial(n+1,2) + 3*binomial(n,2).
%F A000384 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 A000384 a(n) = T(n) + 3*T(n-1), where T(n) is the n-th triangular number. - _Vincenzo Librandi_, Nov 10 2010
%F A000384 a(n) = a(n-1) + 4*n - 3 (with a(0)=0). - _Vincenzo Librandi_, Nov 20 2010
%F A000384 a(n) = A007606(A000290(n)). - _Reinhard Zumkeller_, Feb 12 2011
%F A000384 a(n) = 2*a(n-1) - a(n-2) + 4. - _Ant King_, Aug 26 2011
%F A000384 a(n+1) = A045896(2*n). - _Reinhard Zumkeller_, Dec 12 2011
%F A000384 a(2^n) = 2^(2n+1) - 2^n. - _Ivan N. Ianakiev_, Apr 13 2013
%F A000384 a(n) = binomial(2*n,2). - _Gary Detlefs_, Jul 28 2013
%F A000384 a(n+1) = A128918(2*n+1). - _Reinhard Zumkeller_, Oct 13 2013
%F A000384 a(4*a(n)+7*n+1) =  a(4*a(n)+7*n) + a(4*n+1). - _Vladimir Shevelev_, Jan 24 2014
%F A000384 Sum_{n>=1} 1/a(n) = 2*log(2) = 1.38629436111989...= A016627. - _Vaclav Kotesovec_, Apr 27 2016
%F A000384 Sum_{n>=1} (-1)^n/a(n) = log(2) - Pi/2. - _Vaclav Kotesovec_, Apr 20 2018
%F A000384 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 A000384 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 A000384 a(n) = (A005408(n-1) + A046092(n-1) + A001844(n-1))/2. - _Ralf Steiner_, Feb 27 2020
%F A000384 Product_{n>=2} (1 - 1/a(n)) = 2/3. - _Amiram Eldar_, Jan 21 2021
%F A000384 a(n) = floor(Sum_{k=(n-1)^2..n^2} sqrt(k)), for n >= 1. - _Amrit Awasthi_, Jun 13 2021
%F A000384 a(n+1) = A084265(2*n), n>=0. - _Hartmut F. W. Hoft_, Feb 02 2022
%F A000384 a(n) = A000290(n) + A002378(n-1). - _Charles Kusniec_, Sep 11 2022
%p A000384 A000384:=n->n*(2*n-1); seq(A000384(k), k=0..100); # _Wesley Ivan Hurt_, Sep 27 2013
%t A000384 Table[n*(2 n - 1), {n, 0, 100}] (* _Wesley Ivan Hurt_, Sep 27 2013 *)
%t A000384 LinearRecurrence[{3, -3, 1}, {0, 1, 6}, 50] (* _Harvey P. Dale_, Sep 10 2015 *)
%t A000384 Join[{0}, Accumulate[Range[1, 312, 4]]] (* _Harvey P. Dale_, Mar 26 2016 *)
%t A000384 (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[6], n], {n, 0, 48}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t A000384 PolygonalNumber[6, Range[0, 20]] (* _Eric W. Weisstein_, Aug 17 2017 *)
%t A000384 CoefficientList[Series[x*(1 + 3*x)/(1 - x)^3 , {x, 0, 100}], x] (* _Stefano Spezia_, Sep 02 2018 *)
%o A000384 (PARI) a(n)=n*(2*n-1)
%o A000384 (PARI) a(n) = binomial(2*n,2) \\ _Altug Alkan_, Oct 06 2015
%o A000384 (Haskell)
%o A000384 a000384 n = n * (2 * n - 1)
%o A000384 a000384_list = scanl (+) 0 a016813_list
%o A000384 -- _Reinhard Zumkeller_, Dec 16 2012
%o A000384 (Python) # Intended to compute the initial segment of the sequence, not isolated terms.
%o A000384 def aList():
%o A000384      x, y = 1, 1
%o A000384      yield 0
%o A000384      while True:
%o A000384          yield x
%o A000384          x, y = x + y + 4, y + 4
%o A000384 A000384 = aList()
%o A000384 print([next(A000384) for i in range(49)]) # _Peter Luschny_, Aug 04 2019
%Y A000384 Cf. A000217, A014105, A127672, A027907, A005408, A046092, A001844.
%Y A000384 a(n)= A093561(n+1, 2), (4, 1)-Pascal column.
%Y A000384 a(n) = A100345(n, n-1) for n>0.
%Y A000384 Cf. A002939 (twice a(n): sums of Pythagorean triples (X, Y, Z=Y+1)).
%Y A000384 Cf. A280851.
%K A000384 nonn,easy,nice
%O A000384 0,3
%A A000384 _N. J. A. Sloane_
%E A000384 Partially edited by _Joerg Arndt_, Mar 11 2010