|
%I M3818 N1562 #477 May 11 2024 21:30:48
%S 0,1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425,477,532,
%T 590,651,715,782,852,925,1001,1080,1162,1247,1335,1426,1520,1617,1717,
%U 1820,1926,2035,2147,2262,2380,2501,2625,2752,2882,3015,3151
%N Pentagonal numbers: a(n) = n*(3*n-1)/2.
%C The average of the first n (n > 0) pentagonal numbers is the n-th triangular number. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
%C a(n) is the sum of n integers starting from n, i.e., 1, 2 + 3, 3 + 4 + 5, 4 + 5 + 6 + 7, etc. - _Jon Perry_, Jan 15 2004
%C Partial sums of 1, 4, 7, 10, 13, 16, ... (1 mod 3), a(2k) = k(6k-1), a(2k-1) = (2k-1)(3k-2). - _Jon Perry_, Sep 10 2004
%C Starting with offset 1 = binomial transform of [1, 4, 3, 0, 0, 0, ...]. Also, A004736 * [1, 3, 3, 3, ...]. - _Gary W. Adamson_, Oct 25 2007
%C If Y is a 3-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Nov 23 2007
%C Solutions to the duplication formula 2*a(n) = a(k) are given by the index pairs (n, k) = (5,7), (5577, 7887), (6435661, 9101399), etc. The indices are integer solutions to the pair of equations 2(6n-1)^2 = 1 + y^2, k = (1+y)/6, so these n can be generated from the subset of numbers [1+A001653(i)]/6, any i, where these are integers, confined to the cases where the associated k=[1+A002315(i)]/6 are also integers. - _R. J. Mathar_, Feb 01 2008
%C a(n) is a binomial coefficient C(n,4) (A000332) if and only if n is a generalized pentagonal number (A001318). Also see A145920. - _Matthew Vandermast_, Oct 28 2008
%C Even octagonal numbers divided by 8. - _Omar E. Pol_, Aug 18 2011
%C Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 12, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 08 2011
%C The hyper-Wiener index of the star-tree with n edges (see A196060, example). - _Emeric Deutsch_, Sep 30 2011
%C More generally the n-th k-gonal number is equal to n + (k-2)*A000217(n-1), n >= 1, k >= 3. In this case k = 5. - _Omar E. Pol_, Apr 06 2013
%C Note that both Euler's pentagonal theorem for the partition numbers and Euler's pentagonal theorem for the sum of divisors refer more exactly to the generalized pentagonal numbers, not this sequence. For more information see A001318, A175003, A238442. - _Omar E. Pol_, Mar 01 2014
%C The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Schuetz and Whieldon link). a(n)= Cat(n,3), so enumerates the number of (n+1)-gon partitions of a (3*(n-1)+2)-gon. Analogous sequences are A100157 (k=4) and A234043 (k=5). - _Tom Copeland_, Oct 05 2014
%C Binomial transform of (0, 1, 3, 0, 0, 0, ...) (A169585 with offset 1) and second partial sum of (0, 1, 3, 3, 3, ...). - _Gary W. Adamson_, Oct 05 2015
%C For n > 0, a(n) is the number of compositions of n+8 into n parts avoiding parts 2 and 3. - _Milan Janjic_, Jan 07 2016
%C a(n) is also the number of edges in the Mycielskian of the complete graph K[n]. Indeed, K[n] has n vertices and n(n-1)/2 edges. Then its Mycielskian has n + 3n(n-1)/2 = n(3n-1)/2. See p. 205 of the West reference. - _Emeric Deutsch_, Nov 04 2016
%C Sum of the numbers from n to 2n-1. - _Wesley Ivan Hurt_, Dec 03 2016
%C Also the number of maximal cliques in the n-Andrásfai graph. - _Eric W. Weisstein_, Dec 01 2017
%C Coefficients in the hypergeometric series identity 1 - 5*(x - 1)/(2*x + 1) + 12*(x - 1)*(x - 2)/((2*x + 1)*(2*x + 2)) - 22*(x - 1)*(x - 2)*(x - 3)/((2*x + 1)*(2*x + 2)*(2*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A002412 and A002418. Column 2 of A103450. - _Peter Bala_, Mar 14 2019
%C A generalization of the Comment dated Apr 10 2003 follows. (k-3)*A000292(n-2) plus the average of the first n (2k-1)-gonal numbers is the n-th k-gonal number. - _Charlie Marion_, Nov 01 2020
%C a(n+1) is the number of Dyck paths of size (3,3n+1); i.e., the number of NE lattice paths from (0,0) to (3,3n+1) which stay above the line connecting these points. - _Harry Richman_, Jul 13 2021
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.
%D Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
%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. 1.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 64.
%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).
%D André Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 186.
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 98-100.
%D Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.
%H Daniel Mondot, <a href="/A000326/b000326.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms by T. D. Noe)
%H George E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
%H S. Barbero, U. Cerruti and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
%H Jordan Bell, <a href="https://arxiv.org/abs/math/0510054">Euler and the pentagonal number theorem</a>, arXiv:math/0510054 [math.HO], 2005-2006.
%H Anicius Manlius Severinus Boethius, <a href="https://archive.org/stream/aniciimanliitor01friegoog#page/n114/mode/2up">De institutione arithmetica libri duo</a>, Book 2, sections 13-14.
%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 Stephan Eberhart, <a href="/A000326/a000326_2.pdf">Letter to N. J. A. Sloane, Jan 06 1978</a>, also scanned copy of Mathematical-Physical Correspondence, No. 22, Christmas 1977.
%H Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 1
%H Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 8.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004, 2009. See p. 8.
%H Leonhard Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H Rodney T. Hansen, <a href="http://www.fq.math.ca/Scanned/8-1/hansen.pdf">Arithmetic of pentagonal numbers</a>, Fib. Quart., 8 (1970), 83-87.
%H Alfred Hoehn, <a href="/A000326/a000326.pdf">Illustration of initial terms of A000326, A005449, A045943, A115067</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=339">Encyclopedia of Combinatorial Structures 339</a>
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H Jangwon Ju and Daejun Kim, <a href="https://arxiv.org/abs/2010.16123">The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma</a>, arXiv:2010.16123 [math.NT], 2020.
%H Bir Kafle, Florian Luca and Alain Togbé, <a href="https://doi.org/10.33039/ami.2020.09.002">Pentagonal and heptagonal repdigits</a>, Annales Mathematicae et Informaticae. pp. 137-145.
%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 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 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H R. P. Loh, A. G. Shannon and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.
%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 Cliff Reiter, <a href="http://hdl.handle.net/10385/2864">Polygonal Numbers and Fifty One Stars</a>, Lafayette College, Easton, PA (2019).
%H Alison Schuetz and Gwyneth Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.
%H W. Sierpiński, <a href="/A000326/a000326_1.pdf">Sur les nombres pentagonaux</a>, Bull. Soc. Royale Sciences Liège, 33 (No. 9-10, 1964), 513-517. [Annotated scanned copy]
%H N. J. A. Sloane, <a href="/A000217/a000217.gif">Illustration of initial terms of A000217, A000290, A000326</a>.
%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/AndrasfaiGraph.html">Andrásfai Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mycielskian">Mycielskian</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number">Pentagonal number</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F Product_{m > 0} (1 - q^m) = Sum_{k} (-1)^k*x^a(k). - _Paul Barry_, Jul 20 2003
%F G.f.: x*(1+2*x)/(1-x)^3.
%F E.g.f.: exp(x)*(x+3*x^2/2).
%F a(n) = n*(3*n-1)/2.
%F a(-n) = A005449(n).
%F a(n) = binomial(3*n, 2)/3. - _Paul Barry_, Jul 20 2003
%F a(n) = A000290(n) + A000217(n-1). - _Lekraj Beedassy_, Jun 07 2004
%F a(0) = 0, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - a(n-2) + 3. - _Miklos Kristof_, Mar 09 2005
%F a(n) = Sum_{k=1..n} (2*n - k). - _Paul Barry_, Aug 19 2005
%F a(n) = 3*A000217(n) - 2*n. - _Lekraj Beedassy_, Sep 26 2006
%F a(n) = A126890(n, n-1) for n > 0. - _Reinhard Zumkeller_, Dec 30 2006
%F a(n) = A049452(n) - A022266(n) = A033991(n) - A005476(n). - _Zerinvary Lajos_, Jun 12 2007
%F Equals A034856(n) + (n - 1)^2. Also equals A051340 * [1,2,3,...]. - _Gary W. Adamson_, Jul 27 2007
%F a(n) = binomial(n+1, 2) + 2*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) = 5. - _Jaume Oliver Lafont_, Dec 02 2008
%F a(n) = a(n-1) + 3*n-2 with n > 0, a(0)=0. - _Vincenzo Librandi_, Nov 20 2010
%F a(n) = A000217(n) + 2*A000217(n-1). - _Vincenzo Librandi_, Nov 20 2010
%F a(n) = A014642(n)/8. - _Omar E. Pol_, Aug 18 2011
%F a(n) = A142150(n) + A191967(n). - _Reinhard Zumkeller_, Jul 07 2012
%F a(n) = (A000290(n) + A000384(n))/2 = (A000217(n) + A000566(n))/2 = A049450(n)/2. - _Omar E. Pol_, Jan 11 2013
%F a(n) = n*A000217(n) - (n-1)*A000217(n-1). - _Bruno Berselli_, Jan 18 2013
%F a(n) = A005449(n) - n. - _Philippe Deléham_, Mar 30 2013
%F From _Oskar Wieland_, Apr 10 2013: (Start)
%F a(n) = a(n+1) - A016777(n),
%F a(n) = a(n+2) - A016969(n),
%F a(n) = a(n+3) - A016777(n)*3 = a(n+3) - A017197(n),
%F a(n) = a(n+4) - A016969(n)*2 = a(n+4) - A017641(n),
%F a(n) = a(n+5) - A016777(n)*5,
%F a(n) = a(n+6) - A016969(n)*3,
%F a(n) = a(n+7) - A016777(n)*7,
%F a(n) = a(n+8) - A016969(n)*4,
%F a(n) = a(n+9) - A016777(n)*9. (End)
%F a(n) = A000217(2n-1) - A000217(n-1), for n > 0. - _Ivan N. Ianakiev_, Apr 17 2013
%F a(n) = A002411(n) - A002411(n-1). - _J. M. Bergot_, Jun 12 2013
%F Sum_{n>=1} a(n)/n! = 2.5*exp(1). - _Richard R. Forberg_, Jul 15 2013
%F a(n) = floor(n/(exp(2/(3*n)) - 1)), for n > 0. - _Richard R. Forberg_, Jul 27 2013
%F From _Vladimir Shevelev_, Jan 24 2014: (Start)
%F a(3*a(n) + 4*n + 1) = a(3*a(n) + 4*n) + a(3*n+1).
%F A generalization. Let {G_k(n)}_(n >= 0) be sequence of k-gonal numbers (k >= 3). Then the following identity holds: G_k((k-2)*G_k(n) + c(k-3)*n + 1) = G_k((k-2)*G_k(n) + c(k-3)*n) + G_k((k-2)*n + 1), where c = A000124. (End)
%F A242357(a(n)) = 1 for n > 0. - _Reinhard Zumkeller_, May 11 2014
%F Sum_{n>=1} 1/a(n)= (1/3)*(9*log(3) - sqrt(3)*Pi). - _Enrique Pérez Herrero_, Dec 02 2014. See the decimal expansion A244641.
%F a(n) = (A000292(6*n+k-1)-A000292(k))/(6*n-1)-A000217(3*n+k), for any k >= 0. - _Manfred Arens_, Apr 26 2015 [minor edits from _Wolfdieter Lang_, May 10 2015]
%F a(n) = A258708(3*n-1,1) for n > 0. - _Reinhard Zumkeller_, Jun 23 2015
%F a(n) = A007584(n) - A245301(n-1), for n > 0. - _Manfred Arens_, Jan 31 2016
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi - 6*log(2))/3 = 0.85501000622865446... - _Ilya Gutkovskiy_, Jul 28 2016
%F a(m+n) = a(m) + a(n) + 3*m*n. - _Etienne Dupuis_, Feb 16 2017
%F In general, let P(k,n) be the n-th k-gonal number. Then P(k,m+n) = P(k,m) + (k-2)mn + P(k,n). - _Charlie Marion_, Apr 16 2017
%F a(n) = A023855(2*n-1) - A023855(2*n-2). - _Luc Rousseau_, Feb 24 2018
%F a(n) = binomial(n,2) + n^2. - _Pedro Caceres_, Jul 28 2019
%F Product_{n>=2} (1 - 1/a(n)) = 3/5. - _Amiram Eldar_, Jan 21 2021
%F (n+1)*(a(n^2) + a(n^2+1) + ... + a(n^2+n)) = n*(a(n^2+n+1) + ... + a(n^2+2n)). - _Charlie Marion_, Apr 28 2024
%e Illustration of initial terms:
%e .
%e . o
%e . o o
%e . o o o o
%e . o o o o o o
%e . o o o o o o o o o
%e . o o o o o o o o o o o
%e . o o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o
%e . o o o o o o o o o o o o o o o
%e .
%e . 1 5 12 22 35
%e - _Philippe Deléham_, Mar 30 2013
%p A000326 := n->n*(3*n-1)/2: seq(A000326(n), n=0..100);
%p A000326:=-(1+2*z)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+3 od: seq(a[n], n=0..50); # _Miklos Kristof_, _Zerinvary Lajos_, Feb 18 2008
%t Table[n (3 n - 1)/2, {n, 0, 60}] (* _Stefan Steinerberger_, Apr 01 2006 *)
%t Array[# (3 # - 1)/2 &, 47, 0] (* _Zerinvary Lajos_, Jul 10 2009 *)
%t LinearRecurrence[{3, -3, 1}, {0, 1, 5}, 61] (* _Harvey P. Dale_, Dec 27 2011 *)
%t pentQ[n_] := IntegerQ[(1 + Sqrt[24 n + 1])/6]; pentQ[0] = True; Select[Range[0, 3200], pentQ@# &] (* _Robert G. Wilson v_, Mar 31 2014 *)
%t Join[{0}, Accumulate[Range[1, 312, 3]]] (* _Harvey P. Dale_, Mar 26 2016 *)
%t (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[5], n], {n, 0, 46}] (* _Arkadiusz Wesolowski_, Aug 27 2016 *)
%t CoefficientList[Series[x (-1 - 2 x)/(-1 + x)^3, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 01 2017 *)
%t PolygonalNumber[5, Range[0, 20]] (* _Eric W. Weisstein_, Dec 01 2017 *)
%o (PARI) a(n)=n*(3*n-1)/2
%o (PARI) vector(100, n, n--; binomial(3*n, 2)/3) \\ _Altug Alkan_, Oct 06 2015
%o (PARI) is_a000326 = my(s); n==0 || (issquare (24*n+1, &s) && s%6==5) \\ _Hugo Pfoertner_, Aug 03 2023
%o (Haskell)
%o a000326 n = n * (3 * n - 1) `div` 2 -- _Reinhard Zumkeller_, Jul 07 2012
%o (Magma) [n*(3*n-1)/2 : n in [0..100]]; // _Wesley Ivan Hurt_, Oct 15 2015
%o (GAP) List([0..50],n->n*(3*n-1)/2); # _Muniru A Asiru_, Mar 18 2019
%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 + 3, y + 3
%o A000326 = aList()
%o print([next(A000326) for i in range(47)]) # _Peter Luschny_, Aug 04 2019
%Y The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
%Y Cf. A001318 (generalized pentagonal numbers), A049452, A033570, A010815, A034856, A051340, A004736, A033568, A049453, A002411 (partial sums), A033579.
%Y See 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.
%Y Cf. A240137: sum of n consecutive cubes starting from n^3.
%Y Cf. similar sequences listed in A022288.
%Y Partial sums of A016777.
%K core,nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Incorrect example removed by _Joerg Arndt_, Mar 11 2010
| 