%I A000326 M3818 N1562
%S A000326 0,1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425,477,532,
%T A000326 590,651,715,782,852,925,1001,1080,1162,1247,1335,1426,1520,1617,
%U A000326 1717,1820,1926,2035,2147,2262,2380,2501,2625,2752,2882,3015,3151
%N A000326 Pentagonal numbers: n(3n-1)/2.
%C A000326 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 A000326 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 (perry(AT)globalnet.co.uk), Sep 10 2004
%C A000326 a(n) = A126890(n,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 30 2006
%C A000326 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 R. Janjic (agnus(AT)blic.net), Nov 23 2007
%C A000326 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 (mathar(AT)strw.leidenuniv.nl), 
               Feb 01 2008
%C A000326 a(n) is a binomial coefficient C (n, 4) (A000332) if and only if n is 
               a generalized pentagonal number (A001318). Also see A145920. [From 
               Matthew Vandermast (ghodges14(AT)comcast.net), Oct 28 2008]
%C A000326 Let P(n) = pentagonal number, T(n) = triangular number, then P(n)= T(n)+2*T(n-1) 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 28 2009]
%D A000326 G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 
               44 (No. 4, 2007), 561-573.
%D A000326 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, pages 2 and 311.
%D A000326 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. 
               Soc., 1963; p. 129.
%D A000326 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 189.
%D A000326 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 A000326 R. T. Hansen, Arithmetic of pentagonal numbers, Fib. Quart., 8 (1970), 
               83-87.
%D A000326 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 284.
%D A000326 Clark Kimberling, Complementary Equations, Journal of Integer Sequences, 
               Vol. 10 (2007), Article 07.1.4.
%D A000326 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 
               64.
%D A000326 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000326 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000326 A. Weil, Number theory: an approach through history; from Hammurapi to 
               Legendre, Birkhaeuser, Boston, 1984; see p. 186.
%D A000326 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, 
               pp. 98-100 Penguin Books 1987.
%H A000326 T. D. Noe, <a href="b000326.txt">Table of n, a(n) for n = 0..1000</a>
%H A000326 J. Bell, <a href="http://arXiv.org/abs/math.HO/0510054">Euler and the 
               pentagonal number theorem</a>
%H A000326 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">
               De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 1
%H A000326 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">
               Observatio de summis divisorum</a> p. 8.
%H A000326 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation 
               on the sums of divisors</a> p. 8.
%H A000326 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable 
               properties of the pentagonal numbers</a>
%H A000326 Alfred Hoehn, <a href="a000326.jpg">Illustration of initial terms of 
               A000326, A005449, A045943, A115067</a>
%H A000326 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=339">
               Encyclopedia of Combinatorial Structures 339</a>
%H A000326 Hyun Kwang Kim, <a href="http://com2mac.postech.ac.kr/papers/2001/01-22.pdf">
               On Regular Polytope Numbers</a>
%H A000326 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">Partial Complements and Transposable Dispersions</
               a>, J. Integer Seqs., Vol. 7, 2004.
%H A000326 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000326 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000326 N. J. A. Sloane, <a href="a217.gif">Illustration of initial terms of 
               A000217, A000290, A000326</a>
%H A000326 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PentagonalNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000326 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000326 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A000326 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A000326 Product_{m>0} (1-q^m) = Sum_{k} (-1)^k*x^a(k). - Paul Barry (pbarry(AT)wit.ie), 
               Jul 20 2003
%F A000326 G.f.: x(1+2x)/(1-x)^3. E.g.f.: exp(x)(x+3x^2/2). a(n) = n(3n-1)/2. a(-n) 
               = A005449(n).
%F A000326 a(n) = binomial(3n, 2)/3 - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003
%F A000326 a(n) is the sum of n integers from n, i.e. 1, 2+3, 3+4+5, 4+5+6+7, etc. 
               - Jon Perry (perry(AT)globalnet.co.uk), Jan 15 2004
%F A000326 a(n) = A000290(n) + A000217(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Jun 07 2004
%F A000326 a(0) = 0, a(1) = 1, a(n) = 2*a(n-1)-a(n-2)+3 - Miklos Kristof (kristmikl(AT)freemail.hu), 
               Mar 09 2005
%F A000326 a(n) = sum{k=1..n, 2n-k}; - Paul Barry (pbarry(AT)wit.ie), Aug 19 2005
%F A000326 a(n) = 3*A000217(n) - 2*n . - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Sep 26 2006
%F A000326 a(n)=A049452(n)-A022266(n), example: 70=287-217, etc... a(n)=A033991(n)-A005476(n), 
               example:22=60-38, etc... - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 12 2007
%F A000326 Equals A034856(n) + (n - 1)^2. Also equals A051340 * [1,2,3,...]. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007
%F A000326 Starting with offset 1 = binomial transform of [1, 4, 3, 0, 0, 0,...]. 
               Also, A004736 * [1, 3, 3, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 25 2007
%F A000326 a(n) = C(n+1,2) + 2 C(n,2)
%F A000326 a(n)=A000290(n)+A000217(n-1) (36+15=51 etc...) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 18 2008
%F A000326 a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=0, a(1)=1, a(2)=5 [From Jaume Oliver 
               Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
%F A000326 a(n)=3*n+a(n-1)-5 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A000326 For n=2, a(2)=3*2+0-5=1; n=3, a(3)=3*3+1-5=5; n=4, a(4)=3*4+5-5=12 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A000326 A000326 := n->n*(3*n-1)/2;
%p A000326 A000326:=-(1+2*z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A000326 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); #author:Miklos Kristof - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 18 2008
%t A000326 Table[n(3n - 1)/2, {n, 0, 60}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%t A000326 s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,1,5!,3}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Apr 02 2009]
%t A000326 Array[ #*(3*# - 1)/2 &, 47, 0] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 10 2009]
%t A000326 Table[Sum[i + n - 3, {i, 2, n}], {n, 1, 50}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 11 2009]
%o A000326 (PARI) a(n)=n*(3*n-1)/2
%Y A000326 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 A000326 Cf. A001318 (generalized pentagonal numbers), A005449, A049050, A033570, 
               A010815.
%Y A000326 Cf. A034856, A051340, A004736, A000217, A000290, A000384.
%Y A000326 Sequence in context: A131976 A074376 A134340 this_sequence A022795 A025734 
               A153818
%Y A000326 Adjacent sequences: A000323 A000324 A000325 this_sequence A000327 A000328 
               A000329
%K A000326 core,nonn,easy,nice,new
%O A000326 0,3
%A A000326 N. J. A. Sloane (njas(AT)research.att.com).