A000326 - OEIS

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

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

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

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

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

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

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

Even octagonal numbers divided by 8. - Omar E. Pol, Aug 18 2011

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

The hyper-Wiener index of the star-tree with n edges (see A196060, example). - Emeric Deutsch, Sep 30 2011

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

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

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

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

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

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

Sum of the numbers from n to 2n-1. - Wesley Ivan Hurt, Dec 03 2016

Also the number of maximal cliques in the n-Andrásfai graph. - Eric W. Weisstein, Dec 01 2017

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

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

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

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

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

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

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.

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 291.

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

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).

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

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

Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.

Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

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

W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Royale Sciences Liège, 33 (No. 9-10, 1964), 513-517. [Annotated scanned copy]

Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).

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

G.f.: x*(1+2*x)/(1-x)^3.

E.g.f.: exp(x)*(x+3*x^2/2).

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

a(n) = binomial(3*n, 2)/3. - Paul Barry, Jul 20 2003

a(0) = 0, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - a(n-2) + 3. - Miklos Kristof, Mar 09 2005

a(n) = Sum_{k=1..n} (2*n - k). - Paul Barry, Aug 19 2005

a(n) = binomial(n+1, 2) + 2*binomial(n, 2).

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

a(n) = a(n-1) + 3*n-2 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 20 2010

a(n) = a(n+5) - A016777(n)*5,

a(n) = a(n+6) - A016969(n)*3,

a(n) = a(n+7) - A016777(n)*7,

a(n) = a(n+8) - A016969(n)*4,

a(n) = a(n+9) - A016777(n)*9. (End)

a(n) = floor(n/(exp(2/(3*n)) - 1)), for n > 0. - Richard R. Forberg, Jul 27 2013

a(3*a(n) + 4*n + 1) = a(3*a(n) + 4*n) + a(3*n+1).

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)

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.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi - 6*log(2))/3 = 0.85501000622865446... - Ilya Gutkovskiy, Jul 28 2016

a(m+n) = a(m) + a(n) + 3*m*n. - Etienne Dupuis, Feb 16 2017

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

a(n) = binomial(n,2) + n^2. - Pedro Caceres, Jul 28 2019

Product_{n>=2} (1 - 1/a(n)) = 3/5. - Amiram Eldar, Jan 21 2021

(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

a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * binomial(k, 2)*binomial(3*n+k-1, 2*k). - Peter Bala, Nov 04 2024

Illustration of initial terms:

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. 1 5 12 22 35

Array[# (3 # - 1)/2 &, 47, 0] (* Zerinvary Lajos, Jul 10 2009 *)

LinearRecurrence[{3, -3, 1}, {0, 1, 5}, 61] (* Harvey P. Dale, Dec 27 2011 *)

pentQ[n_] := IntegerQ[(1 + Sqrt[24 n + 1])/6]; pentQ[0] = True; Select[Range[0, 3200], pentQ@# &] (* Robert G. Wilson v, Mar 31 2014 *)

Join[{0}, Accumulate[Range[1, 312, 3]]] (* Harvey P. Dale, Mar 26 2016 *)

(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[5], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *)

CoefficientList[Series[x (-1 - 2 x)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

PolygonalNumber[5, Range[0, 20]] (* Eric W. Weisstein, Dec 01 2017 *)

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

(PARI) vector(100, n, n--; binomial(3*n, 2)/3) \\ Altug Alkan, Oct 06 2015

(PARI) is_a000326 = my(s); n==0 || (issquare (24*n+1, &s) && s%6==5) \\ Hugo Pfoertner, Aug 03 2023

(Haskell)

(Magma) [n*(3*n-1)/2 : n in [0..100]]; // Wesley Ivan Hurt, Oct 15 2015

(GAP) List([0..50], n->n*(3*n-1)/2); # Muniru A Asiru, Mar 18 2019

(Python) # Intended to compute the initial segment of the sequence, not isolated terms.

def aList():

x, y = 1, 1

yield 0

while True:

yield x

x, y = x + y + 3, y + 3

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.

Cf. A240137: sum of n consecutive cubes starting from n^3.

Cf. similar sequences listed in A022288.

core,nonn,easy,nice,changed

Incorrect example removed by Joerg Arndt, Mar 11 2010