A033568 -id:A033568 - cp's OEIS frontend

A000326 Pentagonal numbers: a(n) = n(3n-1)/2.

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

Offset: 0

N. J. A. Sloane

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
a(n) is the largest sum of n positive integers x_1, ..., x_n such that x_i | x_(i+1)+1 for each 1 <= i <= n, where x_(n+1) = x_1. - Yifan Xie, Feb 21 2025

			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

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.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 38, 40.
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.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 52-53, 129-130, 132.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 7-10.
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.

Daniel Mondot, Table of n, a(n) for n = 0..10000 (first 1000 terms by T. D. Noe)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
S. Barbero, U. Cerruti and N. Murru, Transforming Recurrent Sequences by Using the Binomial and Invert Operators, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
Jordan Bell, Euler and the pentagonal number theorem, arXiv:math/0510054 [math.HO], 2005-2006.
Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 13-14.
Charles K. Cook and Michael R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 33.
Stephan Eberhart, Letter to N. J. A. Sloane, Jan 06 1978, also scanned copy of Mathematical-Physical Correspondence, No. 22, Christmas 1977.
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, arXiv:math/0411587 [math.HO], 2004, 2009. See p. 8.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
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 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Jangwon Ju and Daejun Kim, The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma, arXiv:2010.16123 [math.NT], 2020.
Bir Kafle, Florian Luca and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae. pp. 137-145.
Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
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.
R. P. Loh, A. G. Shannon and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567
Cliff Reiter, Polygonal Numbers and Fifty One Stars, Lafayette College, Easton, PA (2019).
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Royale Sciences Liège, 33 (No. 9-10, 1964), 513-517. [Annotated scanned copy]
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Andrásfai Graph.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Pentagonal Number.
Wikipedia, Mycielskian.
Wikipedia, Pentagonal number
Index entries for "core" sequences
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences

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), A049452, A033570, A010815, A034856, A051340, A004736, A033568, A049453, A002411 (partial sums), A033579.
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.
Partial sums of A016777.

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

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

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

A000326 := n->n*(3*n-1)/2: seq(A000326(n), n=0..100);
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

Table[n (3 n - 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 *)
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
```

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

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

# 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
A000326 = aList()
print([next(A000326) for i in range(47)]) # Peter Luschny, Aug 04 2019

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} (2*n - k). - Paul Barry, Aug 19 2005
a(n) = 3*A000217(n) - 2*n. - Lekraj Beedassy, Sep 26 2006
a(n) = A126890(n, n-1) for n > 0. - Reinhard Zumkeller, Dec 30 2006
a(n) = A049452(n) - A022266(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) = 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) = A000217(n) + 2*A000217(n-1). - 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)
A242357(a(n)) = 1 for n > 0. - Reinhard Zumkeller, May 11 2014
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.
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]
a(n) = A258708(3*n-1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
a(n) = A007584(n) - A245301(n-1), for n > 0. - Manfred Arens, Jan 31 2016
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) = A023855(2*n-1) - A023855(2*n-2). - Luc Rousseau, Feb 24 2018
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

Incorrect example removed by Joerg Arndt, Mar 11 2010

A001318 Generalized pentagonal numbers: m(3m - 1)/2, m = 0, +-1, +-2, +-3, ....

Original entry on oeis.org

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335

Offset: 0

N. J. A. Sloane

Partial sums of A026741. - Jud McCranie; corrected by Omar E. Pol, Jul 05 2012
From R. K. Guy, Dec 28 2005: (Start)
"Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):
0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
.....-.-.....+..+.....-..-.....+..+......-...-.......+....
"and you get the pentagonal numbers in pairs, one of positive rank and the other negative.
"Append signs according as the pair have the same (+) or opposite (-) parity.
"Then Euler's pentagonal number theorem is easy to remember:
"p(n-0) - p(n-1) - p(n-2) + p(n-5) + p(n-7) - p(n-12) - p(n-15) ++-- = 0^n
where p(n) is the partition function, the left side terminates before the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 0.
"E.g. p(0) = 1, p(7) = p(7-1) + p(7-2) - p(7-5) - p(7-7) + 0^7 = 11 + 7 - 2 - 1 + 0 = 15."
(End)
The sequence may be used in order to compute sigma(n), as described in Euler's article. - Thomas Baruchel, Nov 19 2003
Number of levels in the partitions of n + 1 with parts in {1,2}.
a(n) is the number of 3 X 3 matrices (symmetrical about each diagonal) M = {{a, b, c}, {b, d, b}, {c, b, a}} such that a + b + c = b + d + b = n + 2, a,b,c,d natural numbers; example: a(3) = 5 because (a,b,c,d) = (2,2,1,1), (1,2,2,1), (1,1,3,3), (3,1,1,3), (2,1,2,3). - Philippe Deléham, Apr 11 2007
Also numbers a(n) such that 24*a(n) + 1 = (6*m - 1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361, ..., m = 0, +-1, +-2, ... . - Zak Seidov, Mar 08 2008
From Matthew Vandermast, Oct 28 2008: (Start)
Numbers n for which A000326(n) is a member of A000332. Cf. A145920.
This sequence contains all members of A000332 and all nonnegative members of A145919. For values of n such that n*(3*n - 1)/2 belongs to A000332, see A145919. (End)
Starting with offset 1 = row sums of triangle A168258. - Gary W. Adamson, Nov 21 2009
Starting with offset 1 = Triangle A101688 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009
Starting with offset 1 can be considered the first in an infinite set generated from A026741. Refer to the array in A175005. - Gary W. Adamson, Apr 03 2010
Vertex number of a square spiral whose edges have length A026741. The two axes of the spiral forming an "X" are A000326 and A005449. The four semi-axes forming an "X" are A049452, A049453, A033570 and the numbers >= 2 of A033568. - Omar E. Pol, Sep 08 2011
A general formula for the generalized k-gonal numbers is given by n*((k - 2)*n - k + 4)/2, n=0, +-1, +-2, ..., k >= 5. - Omar E. Pol, Sep 15 2011
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x + y. - Clark Kimberling, Jun 04 2012
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - Omar E. Pol, Aug 04 2012
a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 26 2013
Conway's relation mentioned by R. K. Guy is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - Omar E. Pol, Feb 01 2013
Start with the sequence of all 0's. Add n to each value of a(n) and the next n - 1 terms. The result is the generalized pentagonal numbers. - Wesley Ivan Hurt, Nov 03 2014
(6k + 1) | a(4k). (3k + 1) | a(4k+1). (3k + 2) | a(4k+2). (6k + 5) | a(4k+3). - Jon Perry, Nov 04 2014
Enge, Hart and Johansson proved: "Every generalised pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalised pentagonal numbers a, b < c such that c = 2a + b." (see link theorem 5). - Peter Luschny, Aug 26 2016
The Enge, et al. result for c >= 5 also holds for c >= 2 if 0 is included as a generalized pentagonal number. That is, 2 = 2*1 + 0. - Michael Somos, Jun 02 2018
Suggestion for title, where n actually matches the list and b-file: "Generalized pentagonal numbers: k(n)*(3*k(n) - 1)/2, where k(n) = A001057(n) = [0, 1, -1, 2, -2, 3, -3, ...], n >= 0" - Daniel Forgues, Jun 09 2018 & Jun 12 2018
Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - Omar E. Pol, Jul 25 2018
The last digits form a symmetric cycle of length 40 [0, 1, 2, 5, ..., 5, 2, 1, 0], i.e., a(n) == a(n + 40) (mod 10) and a(n) == a(40*k - n - 1) (mod 10), 40*k > n. - Alejandro J. Becerra Jr., Aug 14 2018
Only 2, 5, and 7 are prime. All terms are of the form k*(k+1)/6, where 3 | k or 3 | k+1. For k > 6, the value divisible by 3 must have another factor d > 2, which will remain after the division by 6. - Eric Snyder, Jun 03 2022
8*a(n) is the product of two even numbers one of which is n + n mod 2. - Peter Luschny, Jul 15 2022
a(n) is the dot product of [1, 2, 3, ..., n] and repeat[1, 1/2]. a(5) = 12 = [1, 2, 3, 4, 5] dot [1, 1/2, 1, 1/2, 1] = [1 + 1 + 3 + 2 + 5]. - Gary W. Adamson, Dec 10 2022
Every nonnegative number is the sum of four terms of this sequence [S. Realis]. - N. J. A. Sloane, May 07 2023
From Peter Bala, Jan 06 2025: (Start)
The sequence terms are the exponents in the expansions of the following infinite products:
1) Product_{n >= 1} (1 - s(n)*q^n) = 1 + q + q^2 + q^5 + q^7 + q^12 + q^15 + ..., where s(n) = (-1)^(1 + mod(n+1,3)).
2) Product_{n >= 1} (1 - q^(2*n))*(1 - q^(3*n))^2/((1 - q^n)*(1 - q^(6*n))) = 1 + q + q^2 + q^5 + q^7 + q^12 + q^15 + ....
3) Product_{n >= 1} (1 - q^n)*(1 - q^(4*n))*(1 - q^(6*n))^5/((1 - q^(2*n))*(1 - q^(3*n))*(1 - q^(12*n)))^2 = 1 - q + q^2 - q^5 - q^7 + q^12 - q^15 + q^22 + q^26 - q^35 + ....
4) Product_{n >= 1} (1 - q^(2*n))^13/((1 - (-1)^n*q^n)*(1 - q^(4*n)))^5 = 1 - 5*q + 7*q^2 - 11*q^5 + 13*q^7 - 17*q^12 + 19*q^15 - + .... See Oliver, Theorem 1.1. (End)

			G.f. = x + 2*x^2 + 5*x^3 + 7*x^4 + 12*x^5 + 15*x^6 + 22*x^7 + 26*x^8 + 35*x^9 + ...

Enoch Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.
Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 117.
Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 2nd ed., Wiley, NY, 1966, p. 231.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
G. E. Andrews and J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345 [math.NT], 2014.
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See P(q).
Stephen Eberhart, Letter to N. J. A. Sloane, Jan 19 1978.
John Elias, Illustration of Initial Terms: Generalized Penthexagrams.
John Elias, Illustration: Star Number Fractal.
John Elias, Illustration: Generalized Pentagonals In Generalized-Octagonal-Hexagrams.
Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, Opera Omnia, Series I, Vol. 2 (1751), pp. 241-253.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
Leonhard Euler, Observatio de summis divisorum p. 8.
Leonhard Euler, An observation on the sums of divisors, p. 8, arXiv:math/0411587 [math.HO], 2004.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math., Vol. 3, No. 2 (2008), pp. 76-114.
Silvia Heubach and Toufik Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
Alfred Hoehn, Illustration of initial terms.
Barbara H. Margolius, Permutations with inversions, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.
Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, pdf and jpg.
Johannes W. Meijer and Manuel Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Vol. 4, No. 1 (December 2008), pp. 176-187.
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.
Mircea Merca and Maxie D. Schmidt, New Factor Pairs for Factorizations of Lambert Series Generating Functions, arXiv:1706.02359 [math.CO], 2017. See Remark 2.2.
Mircea Merca, Euler's partition function in terms of 2-adic valuation, Bol. Soc. Mat. Mex. 30, 76 (2024). See p. 3.
Ivan Niven, Formal power series, Amer. Math. Monthly, Vol. 76, No. 8 (1969), pp. 871-889.
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Vladimir Pletser, Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials, arXiv preprint arXiv:1409.7972 [math.NT], 2014.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
S. Realis, Question 271, Nouv. Corresp. Math., 4 (1878) 27-29.
Steven J. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5 (December 2011), pp. 339-350.
André Weil, Two lectures on number theory, past and present, L'Enseign. Math., Vol. XX (1974), pp. 87-110; Oeuvres III, pp. 279-302.
Eric Weisstein's World of Mathematics, Pentagonal numbers, Partition Function P.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem.
Wikipedia, Pentagonal number theorem.
M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung.
Keke Zhang, Generalized Catalan numbers, arXiv:2011.09593 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A080995 (characteristic function), A026741 (first differences), A034828 (partial sums), A165211 (mod 2).
Cf. A000326 (pentagonal numbers), A005449 (second pentagonal numbers), A000217 (triangular numbers).
Indices of nonzero terms of A010815, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge.
Union of A036498 and A036499.
Cf. A153384, A168258, A101688, A174739, A175005.
Cf. A074378, A057569, A057570, A007310.
Sequences of generalized k-gonal numbers: this sequence (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Column 1 of A195152.
Squares in APs: A221671, A221672.
Cf. A054440, A260664, A260672.
Quadrisection: A049453(k), A033570(k), A033568(k+1), A049452(k+1), k >= 0.
Cf. A002620.

a:=[0,1,2,5];; for n in [5..60] do a[n]:=2*a[n-2]-a[n-4]+3; od; a; # Muniru A Asiru, Aug 16 2018

a001318 n = a001318_list !! n
a001318_list = scanl1 (+) a026741_list -- Reinhard Zumkeller, Nov 15 2015

[(6*n^2 + 6*n + 1 - (2*n + 1)*(-1)^n)/16 : n in [0..50]]; // Wesley Ivan Hurt, Nov 03 2014

[(3*n^2 + 2*n + (n mod 2) * (2*n + 1)) div 8: n in [0..70]]; // Vincenzo Librandi, Nov 04 2014

A001318 := -(1+z+z**2)/(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
A001318 := proc(n) (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16 ; end proc: # R. J. Mathar, Mar 27 2011

Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]
Select[Accumulate[Range[0,200]]/3,IntegerQ] (* Harvey P. Dale, Oct 12 2014 *)
CoefficientList[Series[x (1 + x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 04 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,2,5,7},70] (* Harvey P. Dale, Jun 05 2017 *)
a[ n_] := With[{m = Quotient[n + 1, 2]}, m (3 m + (-1)^n) / 2]; (* Michael Somos, Jun 02 2018 *)

{a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8}; /* Michael Somos, Mar 24 2011 */

{a(n) = if( n<0, n = -1-n); polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n)}; /* Michael Somos, Mar 24 2011 */

{a(n) = my(m = (n+1) \ 2); m * (3*m + (-1)^n) / 2}; /* Michael Somos, Jun 02 2018 */

def a(n):
    p = n % 2
    return (n + p)*(3*n + 2 - p) >> 3
print([a(n) for n in range(60)])  # Peter Luschny, Jul 15 2022

def A001318(n): return n*(n+1)-(m:=n>>1)*(m+1)>>1 # Chai Wah Wu, Nov 23 2024

@CachedFunction
def A001318(n):
    if n == 0 : return 0
    inc = n//2 if is_even(n) else n
    return inc + A001318(n-1)
[A001318(n) for n in (0..59)] # Peter Luschny, Oct 13 2012

Euler: Product_{n>=1} (1 - x^n) = Sum_{n=-oo..oo} (-1)^n*x^(n*(3*n - 1)/2).
A080995(a(n)) = 1: complement of A090864; A000009(a(n)) = A051044(n). - Reinhard Zumkeller, Apr 22 2006
Euler transform of length-3 sequence [2, 2, -1]. - Michael Somos, Mar 24 2011
a(-1 - n) = a(n) for all n in Z. a(2*n) = A005449(n). a(2*n - 1) = A000326(n). - Michael Somos, Mar 24 2011. [The extension of the recurrence to negative indices satisfies the signature (1,2,-2,-1,1), but not the definition of the sequence m*(3*m -1)/2, because there is no m such that a(-1) = 0. - Klaus Purath, Jul 07 2021]
a(n) = 3 + 2*a(n-2) - a(n-4). - Ant King, Aug 23 2011
Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - Michael Somos, Mar 24 2011
G.f.: x*(1 + x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = n*(n + 1)/6 when n runs through numbers == 0 or 2 mod 3. - Barry E. Williams
a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n > 2. - Ralf Stephan, Apr 26 2003
Sequence consists of the pentagonal numbers (A000326), followed by A000326(n) + n and then the next pentagonal number. - Jon Perry, Sep 11 2003
a(n) = (6*n^2 + 6*n + 1)/16 - (2*n + 1)*(-1)^n/16; a(n) = A034828(n+1) - A034828(n). - Paul Barry, May 13 2005
a(n) = Sum_{k=1..floor((n+1)/2)} (n - k + 1). - Paul Barry, Sep 07 2005
a(n) = A000217(n) - A000217(floor(n/2)). - Pierre CAMI, Dec 09 2007
If n even a(n) = a(n-1) + n/2 and if n odd a(n) = a(n-1) + n, n >= 2. - Pierre CAMI, Dec 09 2007
a(n)-a(n-1) = A026741(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - Ant King, Sep 26 2011
a(n) = (1/2)*ceiling(n/2)*ceiling((3*n + 1)/2). - Mircea Merca, Jul 13 2012
a(n) = (A008794(n+1) + A000217(n))/2 = A002378(n) - A085787(n). - Omar E. Pol, Jan 12 2013
a(n) = floor((n + 1)/2)*((n + 1) - (1/2)*floor((n + 1)/2) - 1/2). - Wesley Ivan Hurt, Jan 26 2013
From Oskar Wieland, Apr 10 2013: (Start)
a(n) = a(n+1) - A026741(n),
a(n) = a(n+2) - A001651(n),
a(n) = a(n+3) - A184418(n),
a(n) = a(n+4) - A007310(n),
a(n) = a(n+6) - A001651(n)*3 = a(n+6) - A016051(n),
a(n) = a(n+8) - A007310(n)*2 = a(n+8) - A091999(n),
a(n) = a(n+10)- A001651(n)*5 = a(n+10)- A072703(n),
a(n) = a(n+12)- A007310(n)*3,
a(n) = a(n+14)- A001651(n)*7. (End)
a(n) = (A007310(n+1)^2 - 1)/24. - Richard R. Forberg, May 27 2013; corrected by Zak Seidov, Mar 14 2015; further corrected by Jianing Song, Oct 24 2018
a(n) = Sum_{i = ceiling((n+1)/2)..n} i. - Wesley Ivan Hurt, Jun 08 2013
G.f.: x*G(0), where G(k) = 1 + x*(3*k + 4)/(3*k + 2 - x*(3*k + 2)*(3*k^2 + 11*k + 10)/(x*(3*k^2 + 11*k + 10) + (k + 1)*(3*k + 4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
Sum_{n>=1} 1/a(n) = 6 - 2*Pi/sqrt(3). - Vaclav Kotesovec, Oct 05 2016
a(n) = Sum_{i=1..n} numerator(i/2) = Sum_{i=1..n} denominator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = A000292(A001651(n))/A001651(n), for n>0. - Ivan N. Ianakiev, May 08 2018
a(n) = ((-5 + (-1)^n - 6n)*(-1 + (-1)^n - 6n))/96. - José de Jesús Camacho Medina, Jun 12 2018
a(n) = Sum_{k=1..n} k/gcd(k,2). - Pedro Caceres, Apr 23 2019
Quadrisection. For r = 0,1,2,3: a(r + 4*k) = 6*k^2 + sqrt(24*a(r) + 1)*k + a(r), for k >= 1, with inputs (k = 0) {0,1,2,5}. These are the sequences A049453(k), A033570(k), A033568(k+1), A049452(k+1), for k >= 0, respectively. - Wolfdieter Lang, Feb 12 2021
a(n) = a(n-4) + sqrt(24*a(n-2) + 1), n >= 4. - Klaus Purath, Jul 07 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(log(3)-1). - Amiram Eldar, Feb 28 2022
a(n) = A002620(n) + A008805(n-1). Gary W. Adamson, Dec 10 2022
E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Aug 01 2024

A005449 Second pentagonal numbers: a(n) = n(3n + 1)/2.

Original entry on oeis.org

0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495, 551, 610, 672, 737, 805, 876, 950, 1027, 1107, 1190, 1276, 1365, 1457, 1552, 1650, 1751, 1855, 1962, 2072, 2185, 2301, 2420, 2542, 2667, 2795, 2926, 3060, 3197, 3337, 3480

Offset: 0

N. J. A. Sloane

Number of edges in the join of the complete graph and the cycle graph, both of order n, K_n * C_n. - Roberto E. Martinez II, Jan 07 2002
Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth, Aug 11 2002
The modular form Delta(q) = q*Product_{n>=1} (1-q^n)^24 = q*(1 + Sum_{n>=1} (-1)^n*(q^(n*(3*n-1)/2)+q^(n*(3*n+1)/2)))^24 = q*(1 + Sum_{n>=1} A033999(n)*(q^A000326(n)+q^a(n)))^24. - Jonathan Vos Post, Mar 15 2006
Row sums of triangle A134403.
Bisection of A001318. - Omar E. Pol, Aug 22 2011
Sequence found by reading the line from 0 in the direction 0, 7, ... and the line from 2 in the direction 2, 15, ... in the square spiral whose vertices are the generalized pentagonal numbers, A001318. - Omar E. Pol, Sep 08 2011
A general formula for the n-th second k-gonal number is given by T(n, k) = n*((k-2)*n+k-4)/2, n>=0, k>=5. - Omar E. Pol, Aug 04 2012
Partial sums give A006002. - Denis Borris, Jan 07 2013
A002260 is the following array A read by antidiagonals:
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
and a(n) is the hook sum: Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013
From Klaus Purath, May 13 2021: (Start)
This sequence and A000326 provide all integers m such that 24*m + 1 is a square. The union of the two sequences is A001318.
If A is a sequence satisfying the recurrence t(n) = 3*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 2 or A(0) = -1, A(1) = n - 1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i + n^2 for i>0. (End)
a(n+1) is the number of Dyck paths of size (3,3n+2), i.e., the number of NE lattice paths from (0,0) to (3,3n+2) which stay above the line connecting these points. - Harry Richman, Jul 13 2021
Binomial transform of [0, 2, 3, 0, 0, 0, ...], being a(n) = 2*binomial(n,1) + 3*binomial(n,2). a(3) = 15 = [0, 2, 3, 0] dot [1, 3, 3, 1] = [0 + 6 + 9 + 0]. - Gary W. Adamson, Dec 17 2022
a(n) is the sum of longest side length of all nondegenerate integer-sided triangles with shortest side length n and middle side length (n + 1), n > 0. - Torlach Rush, Feb 04 2024

			From _Omar E. Pol_, Aug 22 2011: (Start)
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
    O    O O     O O O     O O O O     O O O O O
    -    ---     -----     -------     ---------
    2     7        15         26           40
(End)

Henri Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
A. O. L. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp., Vol. 61, No. 203 (1993), pp. 29-68.
Charles H. Conley and Valentin Ovsienko, Quiddities of polygon dissections and the Conway-Coxeter frieze equation, arXiv:2107.01234 [math.CO], 2021.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 1780: I, pp. 56-75.
Leonhard Euler, Observatio de summis divisorum, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 5, pp. 59-74.
Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067.
D. Suprijanto and I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2211-2217.
Martin Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016789 (first differences), A006002 (partial sums).
Cf. A000217, A000320, A000326, A001318, A033568, A049451, A101165, A101166.
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, this sequence, A045943, A115067, A140090, A140091, A059845, A140672-A140675, A151542.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=3).
Cf. numbers of the form n*((2*k+1)*n+1)/2 listed in A022289 (this sequence is the case k=1).

[n*(3*n + 1) / 2: n in [0..40]]; // Vincenzo Librandi, May 02 2011

A005449:=n->n*(3*n + 1)/2; seq(A005449(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013

Table[n (3 n + 1)/2, {n, 0, 100}] (* Zak Seidov, Jan 31 2012 *)

{a(n) = n * (3*n + 1) / 2} /* Michael Somos, Jul 18 2003 */

a(n) = A110449(n, 1) for n>0.
G.f.: x*(2+x)/(1-x)^3. E.g.f.: exp(x)*(2*x + 3*x^2/2). a(n) = n*(3*n + 1)/2. a(-n) = A000326(n). - Michael Somos, Jul 18 2003
a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard Zumkeller, Dec 03 2004
a(n) = Sum_{j=1..n} n+j. - Zerinvary Lajos, Sep 12 2006
a(n) = A126890(n,n). - Reinhard Zumkeller, Dec 30 2006
a(n) = 2*C(3*n,4)/C(3*n,2), n>=1. - Zerinvary Lajos, Jan 02 2007
a(n) = A000217(n) + A000290(n). - Zak Seidov, Apr 06 2008
a(n) = a(n-1) + 3*n - 1 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A129267(n+5,n). - Philippe Deléham, Dec 21 2011
a(n) = 2*A000217(n) + A000217(n-1). - Philippe Deléham, Mar 25 2013
a(n) = A130518(3*n+1). - Philippe Deléham, Mar 26 2013
a(n) = (12/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 04 2013
a(n) = floor(n/(1-exp(-2/(3*n)))) for n>0. - Richard R. Forberg, Jun 22 2013
a(n) = Sum_{i=1..n} (3*i - 1) for n >= 1. - Wesley Ivan Hurt, Oct 11 2013 [Corrected by Rémi Guillaume, Oct 24 2024]
a(n) = (A000292(6*n+k+1)-A000292(k))/(6*n+1) - A000217(3*n+k+1), for any k >= 0. - Manfred Arens, Apr 26 2015
Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 3*log(3) = 0.89036376976145307522... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(2*n) - A000217(n). - Bruno Berselli, Sep 21 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. - Amiram Eldar, Jan 18 2021
From Klaus Purath, May 13 2021: (Start)
Partial sums of A016789 for n > 0.
a(n) = 3*n^2 - A000326(n).
a(n) = A000326(n) + n.
a(n) = A002378(n) + A000217(n-1) for n >= 1. [Corrected by Rémi Guillaume, Aug 14 2024] (End)
From Klaus Purath, Jul 14 2021: (Start)
b^2 = 24*a(n) + 1 we get by b^2 = (a(n+1) - a(n-1))^2 = (a(2*n)/n)^2.
a(2*n) = n*(a(n+1) - a(n-1)), n > 0.
a(2*n+1) = n*(a(n+1) - a(n)). (End)
A generalization of Lajos' formula, dated Sep 12 2006, follows. Let SP(k,n) = the n-th second k-gonal number. Then SP(2k+1,n) = Sum_{j=1..n} (k-1)*n+j+k-2. - Charlie Marion, Jul 13 2024
a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * binomial(k, 2) * binomial(3*n+k, 2*k). - Peter Bala, Nov 03 2024
For integer m, (6*m + 1)^2*a(n) + a(m) = a((6*m+1)*n + m). - Peter Bala, Jan 09 2025

A049452 Pentagonal numbers with even index.

Original entry on oeis.org

0, 5, 22, 51, 92, 145, 210, 287, 376, 477, 590, 715, 852, 1001, 1162, 1335, 1520, 1717, 1926, 2147, 2380, 2625, 2882, 3151, 3432, 3725, 4030, 4347, 4676, 5017, 5370, 5735, 6112, 6501, 6902, 7315, 7740, 8177, 8626, 9087, 9560, 10045, 10542

Offset: 0

Joe Keane (jgk(AT)jgk.org)

If Y is a 3-subset of an (2n+1)-set X then, for n>=4, a(n-1) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 5,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
a(n) is the sum of 2*n consecutive integers starting from 2*n. - Bruno Berselli, Jan 16 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A033570, A049453, A001318, A033568, A185019.

See index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.

seq(n*(6*n-1),n=0..42); # Zerinvary Lajos, Jun 12 2007

Table[n(6n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,5,22},50] (* Harvey P. Dale, Mar 07 2012 *)

a(n)=n*(6*n-1) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(6*n-1).
G.f.: x*(5+7*x)/(1-x)^3.
a(n) = C(6*n,2)/3. - Zerinvary Lajos, Jan 02 2007
a(n) = A001105(n) + A033991(n) = A033428(n) + A049450(n) = A022266(n) + A000326(n). - Zerinvary Lajos, Jun 12 2007
a(n) = 12*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = 4*A000217(n) + A001107(n). - Bruno Berselli, Feb 11 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=22. - Harvey P. Dale, Mar 07 2012
E.g.f.: (6*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - log(2) - 2*sqrt(3)*arccoth(sqrt(3)). (End)

A051866 14-gonal (or tetradecagonal) numbers: a(n) = n(6n-5).

Original entry on oeis.org

0, 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, 949, 1106, 1275, 1456, 1649, 1854, 2071, 2300, 2541, 2794, 3059, 3336, 3625, 3926, 4239, 4564, 4901, 5250, 5611, 5984, 6369, 6766, 7175, 7596, 8029, 8474, 8931, 9400, 9881, 10374

Offset: 0

N. J. A. Sloane, Dec 15 1999

Sequence found by reading the line from 0, in the direction 0, 14, ... and the parallel line from 1, in the direction 1, 39, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818. Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 14, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
After 0, partial sums of A017533. - Bruno Berselli, Sep 11 2013
This is also a star heptagonal number: a(n) = A000566(n) + 7*A000217(n-1). - Luciano Ancora, Mar 30 2015
Starting with offset 1, this is the binomial transform of (1, 13, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 2015

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing, 2012, page 6.

T. D. Noe, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000566, A001318, A017533, A033568, A195818.

A051866 := proc(n) n*(6*n-5) ; end proc: seq(A051866(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

Table[n*(6*n - 5), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

a(n)=n*(6*n-5); \\ Joerg Arndt, Feb 01 2014

G.f.: x*(1+11*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 12*n + a(n-1) - 11, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = A033568(n) - 1. - Omar E. Pol, Jul 18 2012
a(12*a(n)+67*n+1) = a(12*a(n) + 67*n) + a(12*n + 1). - Vladimir Shevelev, Jan 24 2014
From Amiram Eldar, Oct 20 2020: (Start)
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + log(432))/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*sqrt(3)*arccoth(sqrt(3)) - log(2))/5. (End)
Product_{n>=2} (1 - 1/a(n)) = 6/7. - Amiram Eldar, Jan 21 2021
E.g.f.: exp(x)*x*(1 + 6*x). - Stefano Spezia, Jun 08 2021

A049453 Second pentagonal numbers with even index: a(n) = n(6n+1).

Original entry on oeis.org

0, 7, 26, 57, 100, 155, 222, 301, 392, 495, 610, 737, 876, 1027, 1190, 1365, 1552, 1751, 1962, 2185, 2420, 2667, 2926, 3197, 3480, 3775, 4082, 4401, 4732, 5075, 5430, 5797, 6176, 6567, 6970, 7385, 7812, 8251, 8702, 9165, 9640, 10127, 10626, 11137, 11660, 12195

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Number of edges in the join of the complete tripartite graph of order 3n and the cycle graph of order n, K_n,n,n * C_n. - Roberto E. Martinez II, Jan 07 2002
Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
First bisection of A036498. - Bruno Berselli, Nov 25 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001318, A005449, A033568, A033570, A036498, A049452, A185019, A194454.

Cf. A254963 (comment).

seq(binomial(6*n+1,2)/3, n=0..42); # Zerinvary Lajos, Jan 21 2007

s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[n*(6*n + 1), {n,0,50}] (* G. C. Greubel, Jun 07 2017 *)

x='x+O('x^50); concat([0], Vec(x*(7+5*x)/(1-x)^3)) \\ G. C. Greubel, Jun 07 2017

G.f.: x*(7+5*x)/(1-x)^3.
a(n) = 12*n + a(n-1) - 5 with n > 0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Jun 07 2017
From Amiram Eldar, Feb 18 2022: (Start)
Sum_{n>=1} 1/a(n) = 6 - sqrt(3)*Pi/2 - 2*log(2) - 3*log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi + log(2) + sqrt(3)*log(2 + sqrt(3)) - 6. (End)
E.g.f.: exp(x)*x*(7 + 6*x). - Elmo R. Oliveira, Dec 12 2024

A033570 Pentagonal numbers with odd index: a(n) = (2n+1)(3*n+1).

Original entry on oeis.org

1, 12, 35, 70, 117, 176, 247, 330, 425, 532, 651, 782, 925, 1080, 1247, 1426, 1617, 1820, 2035, 2262, 2501, 2752, 3015, 3290, 3577, 3876, 4187, 4510, 4845, 5192, 5551, 5922, 6305, 6700, 7107, 7526, 7957, 8400, 8855, 9322, 9801, 10292, 10795, 11310, 11837

Offset: 0

N. J. A. Sloane

If Y is a 3-subset of an 2*n-set X then, for n >= 4, a(n-2) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
Sequence found by reading the line (one of the diagonal axes) 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
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that 2 <= x/(n*y) < 3 is given by n/a(n) (for n>1). - Andres Cicuttin, Dec 11 2016
a(n) is the sum of 2*n+1 consecutive integers starting from 2*n+1. - Bruno Berselli, Jan 16 2018

Michael De Vlieger, Table of n, a(n) for n = 0..10000
John Elias, Illustration: Natural number stars.
Leo Tavares, Illustration: Square Block Triangles
Eric Weisstein's World of Mathematics, Pentagonal Number.
Wikipedia, Pentagonal number.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A001318, A033568, A049452, A049453, A191967.

Cf. A016754, A014105.

List([0..50], n-> (2*n+1)*(3*n+1)); # G. C. Greubel, Oct 12 2019

[(2*n+1)*(3*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Dec 11 2016

A033570:=n->(2*n+1)*(3*n+1); seq(A033570(n), n=0..40); # Wesley Ivan Hurt, Mar 18 2014

LinearRecurrence[{3,-3,1},{1,12,35},50]
Table[(2 n + 1) (3 n + 1), {n, 0, 50}] (* or *)
CoefficientList[Series[(1 + 9 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Dec 12 2016 *)
PolygonalNumber[5,Range[1,101,2]] (* Harvey P. Dale, Aug 02 2021 *)

a(n)=(2*n+1)*(3*n+1) \\ Charles R Greathouse IV, Jun 11 2015

[(2*n+1)*(3*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

G.f.: (1 + 9*x + 2*x^2)/(1-x)^3.
a(n) = a(n-1) + 12*n-1 for n > 0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = A000326(2*n+1) = A191967(2*n+1). - Reinhard Zumkeller, Jul 07 2012
a(n) = Sum_{i=1..2*(n+1)-1} 4*(n+1) - 2 - i. - Wesley Ivan Hurt, Mar 18 2014
E.g.f.: (1 + 11*x + 6*x^2)*exp(x). - G. C. Greubel, Oct 12 2019
From Amiram Eldar, Feb 20 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi/(2*sqrt(3)) - 2*log(2) + 3*log(3)/2.
Sum_{n>=0} (-1)^n/a(n) = (1/sqrt(3) - 1/2)*Pi + log(2). (End)
a(n) = A016754(n) + A014105(n). - Leo Tavares, May 24 2022

More terms from Ray Chandler, Dec 08 2011

A036499 Numbers of the form k*(k+1)/6 for k = 2 or 3 modulo 6.

Original entry on oeis.org

1, 2, 12, 15, 35, 40, 70, 77, 117, 126, 176, 187, 247, 260, 330, 345, 425, 442, 532, 551, 651, 672, 782, 805, 925, 950, 1080, 1107, 1247, 1276, 1426, 1457, 1617, 1650, 1820, 1855, 2035, 2072, 2262, 2301, 2501, 2542, 2752, 2795, 3015, 3060, 3290, 3337, 3577, 3626

Offset: 1

Wouter Meeussen

Numbers with an odd number of partitions with an extra odd partition; coefficient of z^p in Product_{n >= 1}(1-z^n) has coefficient (-1).
n such that the number of partitions of n into distinct parts with an odd number of parts exceed by 1 the number of partitions of n into distinct parts with an even number of parts. [Euler's 1754/55 pentagonal number theorem, see, e.g., the Freitag-Busam reference (in German). This reference is from Wolfdieter Lang, Jan 18 2016]
In formal power series, A010815=(product(1-x^k),k>0), ranks of coefficients -1. (A001318=ranks of nonzero (1 or -1) in A010815=ranks of odds terms in A000009).
Quasipolynomial of order 2. - Charles R Greathouse IV, Dec 08 2011
Union of A033568 and A033570. - Ray Chandler, Dec 09 2011

Eberhard Freitag and Rolf Busam, Funktionentheorie 1, Springer, Vierte Auflage, 2006, p. 410.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000009, A001318, A010815, A033568, A033570, A036498.

[(3*n*n-5*n+2)/2+(2*n-1)*(n mod 2): n in [1..50]]; // Vincenzo Librandi, Jan 19 2016

seq(seq((6*k+i)*(6*k+i+1)/6,i=2..3),k=0..50); # Robert Israel, Jan 18 2016

Table[ 1/8*(3 + (-1)^k - 6*k)*(1 + (-1)^k - 2*k), {k, 64} ]
LinearRecurrence[{1,2,-2,-1,1},{1,2,12,15,35},50] (* or *)
CoefficientList[Series[(1+x+8x^2+x^3+x^4)/((1-x)^3(1+x)^2),{x,0,100}],x] (* or *)
Table[(2n+1)(3n+{1,2}),{n,0,24}]//Flatten (* Ray Chandler, Dec 09 2011 *)

a(n)=n*(3*n-5)/2+1+n%2*(2*n-1) \\ Charles R Greathouse IV, Dec 08 2011

a(n) = (3*n*n-5*n+2)/2 + (2*n-1)*(n mod 2). - Frank Ellermann, Mar 16 2002
G.f.: (1+x+8*x^2+x^3+x^4)/((1-x)^3*(1+x)^2). - Ray Chandler, Dec 09 2011
Bisection: a(2*k+1) = A001318(1+4*k) = (2*k+1)*(3*k+1) = A033570(k), a(2*(k+1)) = A001318(2+4*k) = (2*k+1)*(3*k+2) = A033568(k+1), k >= 0. - Wolfdieter Lang, Jan 18 2016
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. - Wesley Ivan Hurt, Jan 18 2016
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(3) - 4*log(2). (End)

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Feb 12 2001

Edited by Ray Chandler, Dec 09 2011

A104585 a(n) = (1/2) * ( 3n^2 - n(-1)^n ).

Original entry on oeis.org

0, 2, 5, 15, 22, 40, 51, 77, 92, 126, 145, 187, 210, 260, 287, 345, 376, 442, 477, 551, 590, 672, 715, 805, 852, 950, 1001, 1107, 1162, 1276, 1335, 1457, 1520, 1650, 1717, 1855, 1926, 2072, 2147, 2301, 2380, 2542, 2625, 2795, 2882, 3060, 3151, 3337, 3432, 3626

Offset: 0

Gary W. Adamson, Mar 17 2005

nonn

Previous name was: Pentagonal wave sequence of the second kind.
Even-indexed terms are pentagonal numbers with even index in A000326. Odd-indexed terms are second pentagonal numbers with odd index in A005449.
A104584, pentagonal wave sequence of the first kind; switches odd and even applications and vice versa in A104585. The pentagonal wave triangle, A104586, has A104584 in odd columns and A104585 in even columns.
Integer values of (n+1)(2n+1)/3 in order of appearance. - Wesley Ivan Hurt, Sep 17 2013
Exponents of q in the identity 1 - Sum_{n >= 0} ( q^(3*n+2)*Product_{k = 1..n} (1 - q^(4*k-1)) ) = 1 - q^2 - q^5 + q^15 + q^22 - q^40 - q^51 + + - - .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15  + + - - .... - Peter Bala, Dec 03 2020

			a(5) = 40 = A005449(5), a second pentagonal number.
a(6) = 51 = A000326(6), a pentagonal number.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Pentagonal number theorem.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A005449, A049452, A033570, A049453, A033568, A104586.

I:=[0, 2, 5, 15, 22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Apr 04 2013

Table[(1/2) (3 n^2 - n (-1)^n), {n, 0, 100}] (* Vincenzo Librandi, Apr 04 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,5,15,22},50] (* Harvey P. Dale, Sep 14 2015 *)

a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ). - Ralf Stephan, Nov 13 2010
G.f.: x*(2+3*x+6*x^2+x^3)/(1-x)^3/(1+x)^2. - Colin Barker, Feb 13 2012
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2) - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) - 3*log(3). (End)

More terms from Colin Barker, Feb 13 2012

Better name, using formula from Ralf Stephan, Joerg Arndt, Sep 17 2013

A104584 a(n) = (1/2) * ( 3n^2 + n(-1)^n ).

Original entry on oeis.org

0, 1, 7, 12, 26, 35, 57, 70, 100, 117, 155, 176, 222, 247, 301, 330, 392, 425, 495, 532, 610, 651, 737, 782, 876, 925, 1027, 1080, 1190, 1247, 1365, 1426, 1552, 1617, 1751, 1820, 1962, 2035, 2185, 2262, 2420

Offset: 0

Gary W. Adamson, Mar 17 2005

Previous name was: Pentagonal wave sequence of the first kind.
Odd-indexed terms = A033570, pentagonal numbers with odd index (1, 12, 35, 70, ...). Even-indexed terms = A049453, 2nd pentagonal numbers with even index (0, 7, 26, 57, 100, ...).
Companion sequence A104585 (Pentagonal wave sequence of the second kind), switches odd with even applications and vice versa. The pentagonal wave sequence triangle A104586 has A104584 in odd columns and A104585 in even columns.
Exponents of q in the identity Sum_{n >= 0} ( q^n*Product_{k = 1..n} (1 - q^(4*k-3)) ) = 1 + q - q^7 - q^12 + q^26 + q^35 - - + + .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15  + + - - .... - Peter Bala, Dec 03 2020

			a(5) = 35 = A000326(5).
a(6) = 57 = A005449(6).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Pentagonal number theorem.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A005449, A049453, A033568, A104585, A104586.

I:=[0, 1, 7, 12, 26]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Apr 04 2013

Table[(1/2) (3 n^2 + n (-1)^n), {n, 0, 100}] (* Vincenzo Librandi, Apr 04 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,7,12,26},50] (* Harvey P. Dale, Feb 14 2023 *)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
a(n) = (1/2) * (3*n^2 + n*(-1)^n ). - Ralf Stephan, May 20 2007
G.f. -x*(1+6*x+3*x^2+2*x^3) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 10 2011
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 4*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 3*log(3) - 6. (End)

Better name, using formula from Ralf Stephan, Joerg Arndt, Sep 17 2013

cp's OEIS Frontend

A000326 Pentagonal numbers: a(n) = n*(3*n-1)/2.

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A001318 Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....

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A005449 Second pentagonal numbers: a(n) = n*(3*n + 1)/2.

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A049452 Pentagonal numbers with even index.

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A051866 14-gonal (or tetradecagonal) numbers: a(n) = n*(6*n-5).

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A000326 Pentagonal numbers: a(n) = n(3n-1)/2.

A001318 Generalized pentagonal numbers: m(3m - 1)/2, m = 0, +-1, +-2, +-3, ....

A005449 Second pentagonal numbers: a(n) = n(3n + 1)/2.

A051866 14-gonal (or tetradecagonal) numbers: a(n) = n(6n-5).

A049453 Second pentagonal numbers with even index: a(n) = n(6n+1).

A033570 Pentagonal numbers with odd index: a(n) = (2n+1)(3*n+1).

A104585 a(n) = (1/2) * ( 3n^2 - n(-1)^n ).

A104584 a(n) = (1/2) * ( 3n^2 + n(-1)^n ).