A017641 -id:A017641 - 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

A091999 Numbers that are congruent to {2, 10} mod 12.

Original entry on oeis.org

2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298, 302, 310, 314, 322, 326, 334

Offset: 1

Ray Chandler, Feb 21 2004

Numbers divisible by 2 but not by 3 or 4. - Robert Israel, Apr 24 2015
For n > 1, a(n) is representable as a sum of four but no fewer consecutive nonnegative integers, i.e., 10 = 1 + 2 + 3 + 4, 14 = 2 + 3 + 4 + 5, 22 = 4 + 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016
Essentially the same as A063221. - Omar E. Pol, Aug 16 2023

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

Second row of A092260.
Cf. A109761 (subsequence).
Cf. A002193, A007310, A101263, A138591, A186424.
Cf. A017545, A017641, A063221.

a091999 n = a091999_list !! (n-1)
a091999_list = 2 : 10 : map (+ 12) a091999_list
-- Reinhard Zumkeller, Jan 21 2013

[6*n-3+(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Apr 23 2015

A091999:=n->6*n-3+(-1)^n: seq(A091999(n), n=1..100); # Wesley Ivan Hurt, Apr 23 2015

Flatten[#+{2,10}&/@(12*Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{2,10,14},60] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 6*n - 3 + (-1)^n \\ David Lovler, Jul 16 2022

a(n) = 2*A007310(n).
a(n) = A186424(n) - A186424(n-2), for n > 1.
a(n) = 12*(n-1) - a(n-1), with a(1)=2. - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = a(n-1) + a(n-2) - a(n-3); a(1)=2, a(2)=10, a(3)=14. - Harvey P. Dale, Jun 24 2013
a(n) = 6*n - 3 + (-1)^n. - Wesley Ivan Hurt, Apr 23 2015
E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - Stefano Spezia, May 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)). - Amiram Eldar, Dec 13 2021
E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - David Lovler, Aug 08 2022
a(n) = A063221(n), n > 1. - Omar E. Pol, Aug 15 2023
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2) (A002193).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/12) (A101263). (End)

A017593 a(n) = 12*n + 6.

Original entry on oeis.org

6, 18, 30, 42, 54, 66, 78, 90, 102, 114, 126, 138, 150, 162, 174, 186, 198, 210, 222, 234, 246, 258, 270, 282, 294, 306, 318, 330, 342, 354, 366, 378, 390, 402, 414, 426, 438, 450, 462, 474, 486, 498, 510, 522, 534, 546, 558, 570, 582, 594, 606, 618, 630, 642

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(73).
Continued fraction expansion of tanh(1/6). - Benoit Cloitre, Dec 17 2002
Also solutions to 5^x + 7^x == 11 (mod 13). - Cino Hilliard, May 10 2003
Numbers m such that the sum of the m-th powers of all 2 X 2 matrices over Z/mZ is a nonzero matrix. - José María Grau Ribas, Jan 31 2014
Positive numbers k for which 1/2 + k/4 + k^2/6 is an integer. - Bruno Berselli, Apr 12 2018

P. Fortuny, J. M. Grau, A. M. Oller-Marcén and I. F. Rúa, On power sums of matrices over a finite commutative ring, arXiv:1505.08132 [math.RA], 2015.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005408, A016825, A016945, A017641, A019691, A030133.

12 Range[0, 200] + 6 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
LinearRecurrence[{2, -1}, {6, 18}, 60] (* Harvey P. Dale, Aug 20 2014 *)

a(n)=12*n+6 \\ Charles R Greathouse IV, Sep 24 2015

[i+6 for i in range(645) if gcd(i,12) == 12] # Zerinvary Lajos, May 21 2009

A030133(a(n)) = 9. - Reinhard Zumkeller, Jul 04 2007
a(n) = 24*n - a(n-1) with n > 0, a(0)=6. - Vincenzo Librandi, Nov 19 2010
a(0)=6, a(1)=18; for n > 1, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Aug 20 2014
G.f.: 6*(1+x)/(1-x)^2. - Wolfdieter Lang, Oct 27 2020
Sum_{n>=0} (-1)^n/a(n) = Pi/24 (A019691). - Amiram Eldar, Dec 12 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2) * sin(5*Pi/24).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2) * cos(5*Pi/24). (End)
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 6*exp(x)*(1 + 2*x).
a(n) = 6*A005408(n) = 3*A016825(n) = 2*A016945(n). (End)

Typos in sequence (270 was 2,70 and 510 was 5,10) fixed by Peter Luschny, Dec 14 2009

A030132 Digital root of Fibonacci(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8

Offset: 0

youngelder(AT)webtv.net (Ana)

Any initial pair (a(0), a(1)) of nonzero single-digit numbers enters a cycle of length 24, except for the 8 cases where 3 divides both a(0), a(1) and (a(0), a(1)) != (9, 9), which enter a cycle of length 8 and (9, 9), which is immediately periodic of period length 1. - Jonathan Vos Post, Dec 29 2005 [Corrected by Jianing Song, Apr 17 2021]
First term that differs from A004090 is a(10). In general, all terms of A004090 having one digit are the same in this sequence. - Alonso del Arte, Sep 16 2012
Decimal expansion of 12484270798876404618091 / 1111111111111111111111110 = 0.0[112358437189887641562819] (periodic). - Daniel Forgues, Feb 27 2017

			a(10) = 1 because F(10) = 55, and since 5 + 5 = 10 and 1 + 0 = 1 is the digital root of 55.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Robert Bruce Gray, Another context for the first 48 terms of this sequence, May 08 2025
S. Marivani and others, Digital Roots of Fibonacci Numbers: Problem 10974, Amer. Math. Monthly, 111 (No. 7, 2004), 628.
Colm Mulcahy, Gibonacci Bracelets.
Marc Renault, The Fibonacci sequence modulo m
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000045 (Fibonacci numbers), A010888 (digital roots), A004090, A007953, A030133.

Cf. A001175, A010077, A017641, A065076.

a030132 n = a030132_list !! n
a030132_list =
   0 : 1 : map a007953 (zipWith (+) a030132_list (tail a030132_list))
-- Reinhard Zumkeller, Aug 20 2011

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&, n]; Table[If[n == 0, 0, digitalRoot[Fibonacci[n]]], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
Table[NestWhile[Total[IntegerDigits[#]]&, Fibonacci[n], # > 9 &], {n, 0, 90}] (* Harvey P. Dale, May 07 2012 *)
PadRight[{0},120,{9,1,1,2,3,5,8,4,3,7,1,8,9,8,8,7,6,4,1,5,6,2,8,1}] (* Harvey P. Dale, Jul 20 2024 *)

a(n)=if(n,(fibonacci(n)-1)%9+1,0) \\ Charles R Greathouse IV, Jan 23 2013

a(n + 1) = sum of digits of (a(n) + a(n - 1)).
Periodic with period 24 = A001175(9) given by {1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9}.
a(n) + a(n + 1) = A010077(n + 4); a(A017641(n)) = 9. - Reinhard Zumkeller, Jul 04 2007
G.f.: x*( -1 -x -2*x^2 -3*x^3 -5*x^4 -8*x^5 -4*x^6 -3*x^7 -7*x^8 -x^9 -8*x^10 -9*x^11 -8*x^12 -8*x^13 -7*x^14 -6*x^15 -4*x^16 -x^17 -5*x^18 -6*x^19 -2*x^20 -8*x^21 -x^22 -9*x^23 ) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) *(1+x^4) *(x^8-x^4+1) ). - R. J. Mathar, Feb 08 2013

Entry revised by N. J. A. Sloane, Aug 29 2004

A269100 a(n) = 13*n + 11.

Original entry on oeis.org

11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739

Offset: 0

Bruno Berselli, Feb 19 2016

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 11, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k =  2: A153080,
k =  6: A186113,
k =  7: A269044,
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A094784, A106389.
Cf. A140373.
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), this sequence (q=11).

Magma
```
[13*n+11: n in [0..60]];
```

13 Range[0,60] + 11
Range[11, 800, 13]
Table[13 n + 11, {n, 0, 60}] (* Bruno Berselli, Feb 22 2016 *)
LinearRecurrence[{2,-1},{11,24},60] (* Harvey P. Dale, Jun 14 2023 *)

Maxima
```
makelist(13*n+11, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+11)
```
Python
```
[13*n+11 for n in range(61)]
```
Sage
```
[13*n+11 for n in range(61)]
```

G.f.: (11 + 2*x)/(1 - x)^2.
a(n) = -A153080(-n-1).
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
E.g.f.: (11 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A332516 Numbers k such that phi(k) == 10 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

11, 22, 23, 46, 47, 59, 71, 83, 94, 107, 118, 131, 142, 166, 167, 179, 191, 214, 227, 239, 251, 262, 263, 311, 334, 347, 358, 359, 382, 383, 419, 431, 443, 454, 467, 478, 479, 491, 502, 503, 526, 563, 587, 599, 622, 647, 659, 683, 694, 718, 719, 743, 766, 827

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (3/8) * x/log(x).

			23 is a term since phi(23) = 22 == 10 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A017641, A332511, A332512, A332513, A332514, A332515.

[k:k in [1..850]| EulerPhi(k) mod 12 eq 10]; // Marius A. Burtea, Feb 14 2020

Select[Range[800], Mod[EulerPhi[#], 12] == 10 &]

A337610 Positive integers m such that A126287^k(m) = m for some positive integer k.

Original entry on oeis.org

1, 10, 15, 22, 26, 33, 34, 46, 51, 58, 65, 69, 70, 82, 86, 87, 94, 105, 106, 118, 122, 123, 130, 134, 141, 142, 146, 154, 159, 166, 177, 178, 190, 195, 202, 206, 213, 214, 215, 218, 226, 231, 238, 249, 250, 254, 262, 266, 267, 274, 285, 286, 298, 302, 303, 310

Offset: 1

Ely Golden, Oct 07 2020

A126287^k(m) means apply A126287 to m k times.
Equivalently, the numbers that belong to a cycle under the map x -> A126287(x).
For any term m in this sequence, A126287(A126287(m)) = m.
Supersequence of A017641. Moreover, this sequence (excluding the first term) can be represented as an infinite union of arithmetic progressions.
There are no primes in this sequence.

			10 is a term since A126287(A126287(10)) = A126287(15) = 10.

Ely Golden, Table of n, a(n) for n = 1..10000
Ely Golden, Python program for generating terms of this sequence

Cf. A017641, A126287, A337609, A337611, A337612.

A017643 a(n) = (12n+10)^3.

Original entry on oeis.org

1000, 10648, 39304, 97336, 195112, 343000, 551368, 830584, 1191016, 1643032, 2197000, 2863288, 3652264, 4574296, 5639752, 6859000, 8242408, 9800344, 11543176, 13481272, 15625000, 17984728, 20570824, 23393656, 26463592, 29791000, 33386248, 37259704

Offset: 0

N. J. A. Sloane

6n + 5 = (12n + 10) / 2 is never a square, as 5 is not a quadratic residue modulo 6. Using this, we can show that each term has an even square part and an even squarefree part, neither part being a power of 2. (Less than 2% of integers have this property - see A339245.) - Peter Munn, Dec 14 2020

Eric Weisstein's World of Mathematics, Quadratic Residue.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A000578, A017641 are used in a formula defining this sequence.
Subsequence of A339245.
Cf. A017642, A017644.

A017643:=(12*n+10)^3; seq(A017643(n), n=0..100); # Wesley Ivan Hurt, Nov 25 2013

(12Range[0,30]+10)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{1000,10648,39304,97336},30] (* Harvey P. Dale, Sep 30 2011 *)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1000, a(1)=10648, a(2)=39304, a(3)=97336. [Harvey P. Dale, Sep 30 2011]

a(n) = A017641(n)^3 = A000578(A017641(n)). - Michel Marcus, Nov 25 2013

A140777 a(n) = 2*prime(n) - 4.

Original entry on oeis.org

0, 2, 6, 10, 18, 22, 30, 34, 42, 54, 58, 70, 78, 82, 90, 102, 114, 118, 130, 138, 142, 154, 162, 174, 190, 198, 202, 210, 214, 222, 250, 258, 270, 274, 294, 298, 310, 322, 330, 342, 354, 358, 378, 382, 390, 394, 418, 442, 450, 454, 462, 474, 478, 498, 510, 522

Offset: 1

Leroy Quet, May 29 2008, May 31 2008

A number n is included if (p + n/p) is prime, where p is the smallest prime that divides n. Since all terms of this sequence are even (or otherwise p + n/p would be even and not a prime), p is always 2. So this sequence is the set of all even numbers n where (2 + n/2) is prime.
The entries are also encountered via the bilinear transform approximation to the natural log (unit circle). Specifically, evaluating 2(x-1)/(x+1) at x = 2, 3, 4, ..., the terms of this sequence are seen ahead of each new prime encountered. Additionally, the position of those same primes will occur at the entry positions. For clarity, the evaluation output is 2, 3, 1, 1, 6, 5, 4, 3, 10, 7, 3, 2, 14, 9, 8, 5, 18, 11, ..., where the entries ahead of each new prime are 2, 6, 10, 18, ... . As an aside, the same mechanism links this sequence to A165355. - Bill McEachen, Jan 08 2015
As a follow-up to previous comment, it appears that the numerators and denominators of 2(x-1)/(x+1) are respectively given by A145979 and A060819, but with different offsets. - Michel Marcus, Jan 14 2015
Subset of the union of A017641 & A017593. - Michel Marcus, Sep 01 2020

			The smallest prime dividing 42 is 2. Since 2 + 42/2 = 23 is prime, 42 is included in this sequence.

Cf. A000040, A020639, A040976, A111234, A140775, A140776.
Cf. A060819, A145979, A165355.
Cf. A017641, A017593.

[2*NthPrime(n)-4: n in [1..80]]; // Vincenzo Librandi, Feb 19 2015

A020639 := proc(n) local dvs,p ; dvs := sort(convert(numtheory[divisors](n),list)) ; for p in dvs do if isprime(p) then RETURN(p) ; fi ; od: error("%d",n) ; end: A111234 := proc(n) local p ; p := A020639(n) ; p+n/p ; end: isA140777 := proc(n) RETURN(isprime(A111234(n))) ; end: for n from 2 to 1200 do if isA140777(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, May 31 2008
seq(2*ithprime(i)-4, i=1..1000); # Robert Israel, Jan 09 2015

fQ[n_] := Block[{p = First@ First@ Transpose@ FactorInteger@ n}, PrimeQ[p + n/p] == True]; Select[ Range[2, 533], fQ@# &] (* Robert G. Wilson v, May 30 2008 *)
Table[2 Prime[n] - 4, {n, 60}] (* Vincenzo Librandi, Feb 19 2015 *)

vector(100, n, 2*prime(n) - 4) \\ Michel Marcus, Jan 09 2015

a(n) = 2*A040976(n). - Michel Marcus, Jan 09 2015

More terms from Robert G. Wilson v and R. J. Mathar, May 30 2008

A247477 Largest m such that n/m + n divides (n/m)^(n/m) + n, (n/m)^n + n/m and n^(n/m) + n/m, or 0 if no such m exists.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 4, 1, 6, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2

Offset: 1

Juri-Stepan Gerasimov, Jan 19 2015

nonn

Call m a superdivisor of n if n/m + n divides (n/m)^(n/m) + n, (n/m)^n + n/m and n^(n/m) + n/m. Then a(n) is the largest superdivisor of n, or 0 if n has no superdivisors.
Conjecture: smallest k such that k/m = n and k/m + k divides (k/m)^(k/m) + k, (k/m)^k + k/m, k^(k/m) + k/m, or 0 if no such k exists: 2, 1, 10, 0, 36, 0, 78, 0, 136, 0, 210, 0, 312, 0, 406, 0, ...
Conjecture:
1 = odd superdivisor of 2n + 1 (or A005408(n));
m = even superdivisor of m*(2m + 2)*n + m*(2m + 1).
That is,
2 = even superdivisor of 12n + 10 (or A017641(n)),
4 = even superdivisor of 40n + 36,
...
Smallest n with more than 1 superdivisor is n = 406 with superdivisors {2, 14}. - Michael De Vlieger, Feb 09 2015
Smallest k such that number of superdivisors of k is equal to n: 2, 1, 406, 2926, ... - Juri-Stepan Gerasimov, Feb 12 2015
Conjecture: the superdivisor constant is equal to 1/2 + sum_{n >= 1} 1/(4*A000217(2n)) - Sum_{n >= 1} 1/b(n) - Sum_{n >= 1} 1/c(n) - Sum_{n >= 1} 1/d(n), ... = 0.64.., where b(n) = numbers with 2 superdivisors {or 406, 430, 646, 666, 826, 1090, 1236, 1246, 1378, 1596, 1666, 1750, 2002, 2028, 2346, 2410, 2506, 2782, 2796, 2850, ...), c(n) = numbers with 3 superdivisors {or 2926, ...), d(n) = numbers with 4 superdivisors, ...  - Juri-Stepan Gerasimov, Feb 18 2015
A000027 = A254748 U 1-superdivisor numbers U 2-superdivisor numbers U 3-superdivisor numbers U 4-superdivisor numbers U ... - Juri-Stepan Gerasimov, Feb 19 2015
Let n = k*d with d odd. Then, k is a superdivisor of n iff d^(d-1) == 1 (mod k+1) and d^(k-1) == -1 (mod k+1). (Sometimes the numbers d are called the superdivisors of n, as in A272538 and possibly A254748.) - Charlie Neder, Jun 02 2019

			a(10) = 2 because 10/2 + 10 = 15 divides (10/2)^(10/2) + 10 = 3135, (10/2)^10 + 10/2 = 9765630, 10^(10/2) + 10/2 = 100005, i.e., 3135/15 = 209, 9765630/15 = 651042, 100005/15 = 6667.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000217, A005408, A017641, A254748.

superdivisors[n_] := Select[Range@ n, And[Mod[(n/#)^(n/#) + n, n/# + n] == 0, Mod[(n/#)^n + n/#, n/# + n] == 0, Mod[n^(n/#) + n/#, n/# + n] == 0] &] /. {} -> 0; Min /@ Array[superdivisors, 94] (* Michael De Vlieger, Feb 09 2015 *)

a(n)=fordiv(n,d,my(m=n/d,k=d+n); if(Mod(d,k)^d==-n && Mod(d,k)^n==-d && Mod(n,k)^d==-d, return(m))); 0 \\ Charles R Greathouse IV, Feb 19 2015

cp's OEIS Frontend

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

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A091999 Numbers that are congruent to {2, 10} mod 12.

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A017593 a(n) = 12*n + 6.

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A030132 Digital root of Fibonacci(n).

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A269100 a(n) = 13*n + 11.

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A332516 Numbers k such that phi(k) == 10 (mod 12), where phi is the Euler totient function (A000010).

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