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A000326 Pentagonal numbers: n*(3*n-1)/2.
(Formerly M3818 N1562)
313
0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

a(n) = A126890(n, n-1) for n > 0. - Reinhard Zumkeller, Dec 30 2006

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

Let P(n) = pentagonal number, T(n) = triangular number, then P(n)= T(n) + 2*T(n-1). - Vincenzo Librandi, Nov 20 2010

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 the Euler's pentagonal theorem for the partition numbers and the 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

A242357(a(n)) = 1 for n > 0. - Reinhard Zumkeller, May 11 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

REFERENCES

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.

C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.

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.

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).

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.

LINKS

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

Jordan Bell, Euler and the pentagonal number theorem

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

Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 13-14.

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

Leonhard Euler, Observatio de summis divisorum p. 8.

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

Leonhard Euler, On the remarkable properties of the pentagonal numbers

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

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 339

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

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

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

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567

Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.

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

Eric Weisstein's World of Mathematics, Pentagonal Number

Wikipedia, Pentagonal number

Index entries for "core" sequences

Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).

Index entries for two-way infinite sequences

FORMULA

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) = A005449(n).

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

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

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, 2n - k}. - Paul Barry, Aug 19 2005

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

a(n) = A049452(n) - A022266(n), a(n) = A033991(n) - A005476(n). - Zerinvary Lajos, Jun 12 2007

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

a(n) = C(n+1, 2) + 2 C(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) = A014642(n)/8. - Omar E. Pol, Aug 18 2011

a(n) = A142150(n) + A191967(n). - Reinhard Zumkeller, Jul 07 2012

a(n) = (A000290(n) + A000384(n))/2 = (A000217(n) + A000566(n))/2 = A049450(n)/2. - Omar E. Pol, Jan 11 2013

a(n) = n*A000217(n) - (n-1)*A000217(n-1). - Bruno Berselli, Jan 18 2013

a(n) = A005449(n) - n. - Philippe Deléham, Mar 30 2013

From Oskar Wieland, Apr 10 2013: (Start)

a(n) = a(n+1) - A016777(n),

a(n) = a(n+2) - A016969(n),

a(n) = a(n+3) - A016777(n)*3 = a(n+3) - A017197(n),

a(n) = a(n+4) - A016969(n)*2 = a(n+4) - A017641(n),

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) = A000217(2n-1) - A000217(n-1), for n > 0. - Ivan N. Ianakiev, Apr 17 2013

a(n) = A002411(n) - A002411(n-1). - J. M. Bergot, Jun 12 2013

Sum(n >= 1, a(n)/n! ) = 2.5*exp(1). - Richard R. Forberg, Jul 15 2013

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

From Vladimir Shevelev, Jan 24 2014: (Start)

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)

EXAMPLE

Illustration of initial terms:

.

.                                       o

.                                     o o

.                          o        o o o

.                        o o      o o o o

.                o     o o o    o o o o o

.              o o   o o o o    o o o o o

.        o   o o o   o o o o    o o o o o

.      o o   o o o   o o o o    o o o o o

.  o   o o   o o o   o o o o    o o o o o

.

.  1    5     12       22           35

- Philippe Deléham, Mar 30 2013

MAPLE

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

A000326:=-(1+2*z)/(z-1)**3; # Simon 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); # Miklos Kristof, Zerinvary Lajos, Feb 18 2008

MATHEMATICA

Table[n(3n - 1)/2, {n, 0, 60}] (* Stefan Steinerberger, Apr 01 2006 *)

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 *)

PROG

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

(Haskell)

a000326 n = n * (3 * n - 1) `div` 2  -- Reinhard Zumkeller, Jul 07 2012

CROSSREFS

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, A034856, A051340, A004736, A033568, A049453, A002411 (partial sums), A033579.

n-gonal numbers: A000217, A000290, this sequence, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865-A051876.

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.

Sequence in context: A131976 A074376 A134340 * A022795 A025734 A153818

Adjacent sequences:  A000323 A000324 A000325 * A000327 A000328 A000329

KEYWORD

core,nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Incorrect example removed by Joerg Arndt, Mar 11 2010

Broken link to Hyun Kwang Kim's paper fixed by Felix Fröhlich, Jun 16 2014

STATUS

approved

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Last modified November 28 01:02 EST 2014. Contains 250286 sequences.