0,3

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

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

a(n) = A126890(n,n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

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

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

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]

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]

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.

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. 1.

R. T. Hansen, Arithmetic of pentagonal numbers, Fib. Quart., 8 (1970), 83-87.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 64.

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).

A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 186.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 98-100 Penguin Books 1987.

T. D. Noe, Table of n, a(n) for n = 0..1000

J. Bell, Euler and the pentagonal number theorem

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 1

L. Euler, Observatio de summis divisorum p. 8.

L. Euler, An observation on the sums of divisors p. 8.

L. Euler, On the remarkable properties of the pentagonal numbers

Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 339

Hyun Kwang Kim, On Regular Polytope Numbers

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

S. 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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for "core" sequences

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

Product_{m>0} (1-q^m) = Sum_{k} (-1)^k*x^a(k). - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003

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).

a(n) = binomial(3n, 2)/3 - Paul Barry (pbarry(AT)wit.ie), Jul 20 2003

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

a(n) = A000290(n) + A000217(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004

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

a(n) = sum{k=1..n, 2n-k}; - Paul Barry (pbarry(AT)wit.ie), Aug 19 2005

a(n) = 3*A000217(n) - 2*n . - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 26 2006

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

Equals A034856(n) + (n - 1)^2. Also equals A051340 * [1,2,3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007

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

a(n) = C(n+1,2) + 2 C(n,2)

a(n)=A000290(n)+A000217(n-1) (36+15=51 etc...) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 18 2008

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]

a(n)=3*n+a(n-1)-5 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

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]

A000326 := n->n*(3*n-1)/2;

A000326:=-(1+2*z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

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

Table[n(3n - 1)/2, {n, 0, 60}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

s=0; lst={}; Do[s+=n; AppendTo[lst, s], {n, 1, 5!, 3}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]

Array[ #*(3*# - 1)/2 &, 47, 0] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

Table[Sum[i + n - 3, {i, 2, n}], {n, 1, 50}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]

(PARI) a(n)=n*(3*n-1)/2

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.

Cf. A001318 (generalized pentagonal numbers), A005449, A049050, A033570, A010815.

Cf. A034856, A051340, A004736, A000217, A000290, A000384.

Sequence in context: A131976 A074376 A134340 this_sequence A022795 A025734 A153818

Adjacent sequences: A000323 A000324 A000325 this_sequence A000327 A000328 A000329

core,nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).