cp's OEIS Frontend

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

From Floor van Lamoen, Jul 21 2001: (Start)

Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,2,.... The spiral begins:

.

56--55--54--53--52

/ \

57 33--32--31--30 51

/ / \ \

58 34 16--15--14 29 50

/ / / \ \ \

59 35 17 5---4 13 28 49

/ / / / \ \ \ \

60 36 18 6 0 3 12 27 48

/ / / / / . / / / /

61 37 19 7 1---2 11 26 47

\ \ \ \ . / / /

62 38 20 8---9--10 25 46

\ \ \ . / /

63 39 21--22--23--24 45

\ \ . /

64 40--41--42--43--44

\ .

65--66--67--68--69--70

(End)

Starting with offset 1 = binomial transform of [2, 8, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009

Number of possible pawn moves on an (n+1) X (n+1) chessboard (n=>3). - Johannes W. Meijer, Feb 04 2010

a(n) = A069905(6n-1): Number of partitions of 6*n-1 into 3 parts. - Adi Dani, Jun 04 2011

Even octagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011

Partial sums give A011379. - Omar E. Pol, Jan 12 2013

First differences are A016933; second differences equal 6. - Bob Selcoe, Apr 02 2015

For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-2; {2, 2n-1, 6, 2n-1, 2, 18n-4}]. - Magus K. Chu, Oct 13 2022

X values of solutions to the equation 6*X^3 + X^2 = Y^2. - Mohamed Bouhamida, Nov 06 2007

Arithmetic averages of the k triangular numbers 0, 1, 3, 6, ..., (k-1)*k/2 that take integer values. - Vladimir Joseph Stephan Orlovsky, Aug 05 2009 [edited by Jon E. Schoenfield, Jan 10 2015]

Even terms in A186423; union of A033579 and A033580, A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011

a(n) are integers produced by Sum_{i = 1..k-1} i*(k-i)/k, for some k > 0. Values for k are given by A007310 = sqrt(6*a(n)+1), the square roots of those perfect squares. - Richard R. Forberg, Feb 16 2015

Equivalently, numbers of the form 2*h*(3*h+1), where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... (see also the sixth comment of A152749). - Bruno Berselli, Feb 02 2017

Equivalently, numbers n such that either 6n-1 or 6n+1 is composite (or both are).

Numbers k such that 36*k^2 - 1 is not a product of twin primes. - Artur Jasinski, Dec 12 2007

Apart from initial zero, union of A046953 and A046954. - Reinhard Zumkeller, Jul 13 2014

From Bob Selcoe, Nov 18 2014: (Start)

Complementary sequence to A002822.

For all k >= 1, a(n) are the only positive numbers congruent to the following residue classes:

f == k (mod 6k+-1);

g == (5k-1) (mod 6k-1);

h == (5k+1) (mod 6k+1).

All numbers in classes g and h will be in this sequence; for class f, the quotient must be >= 1.

When determining which numbers are contained in this sequence, it is only necessary to evaluate f, g and h when the moduli are prime and the dividends are >= 2*k*(3*k - 1) (i.e., A033579(k)).

(End)

From Jason Kimberley, Oct 14 2015: (Start)

Numbers n such that A001222(A136017(n)) > 2.

The disjoint union of A060461, A121763, and A121765.

(End)

From Ralf Steiner, Aug 08 2018 (Start)

Conjecture 1: With u(k) = floor(k(k + 1)/4) one has A071538(a(u(k))*6) = a(u(k)) - u(k) + 1, for k >= 2 (u > 1).

Conjecture 2: In the interval [T(k-1)+1, T(k)], with T(k) = A000217(k), k >= 2, there exists at least one number that is not a member of the present sequence. (End)

Also: numbers of the form n*p +- round(p/6) with some positive integer n and prime p >= 5. [Proof available on demand.] - M. F. Hasler, Jun 25 2019

8 times pentagonal numbers. - Omar E. Pol, Dec 11 2008

Sequence found by reading the line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

The sequence forms the even nesting cube-frames (see illustrations in A000567), which separate and appear according to formula along the axes on the zero-centered and one-centered hexagonal number spirals, as well as the axes of the zero-centered and one-centered square number spirals. See illustrations in links. - John Elias, Jul 20 2022

The product of 4 successive terms of an arithmetic progression + square of the common difference is a square: a(n) = the square arising as the sum of first four terms of an arithmetic progression + n^2 where 1 is the first term and n is the common difference. a(1) = 25 = 1*2*3*4+1 a(2) = 121 = 1*3*5*7 +2^2 a(3) = 289 = 1*4*7*10 + 3^2, etc. - Amarnath Murthy, Mar 25 2004

If Y is a fixed 2-subset of a (6n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

Sequence found by reading the line from 25 in the direction 25, 121,... in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, Jul 28 2016

Sequences of numbers of the form n*(n*k - k + 6)/2:

. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;

. k=11: a(n);

. k=12: A094159;

. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;

. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;

. k=15: A152773;

. k=16: A139272;

. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;

. k=18: A152751;

. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;

. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;

. k=21: A152759;

. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;

. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;

. k=24: A152767;

. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;

. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;

. k=27: A153783;

. k=28: A195021;

. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;

. k=30: A153448;

. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;

. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;

. k=33: A153875.

Also:

a(n) - n = A180223(n);

a(n) + n = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;

a(n) - 2*n = A051865(n);

a(n) + 2*n = A022268(n);

a(n) - 3*n = A152740(n-1);

a(n) + 3*n = A022269(n);

a(n) - 4*n = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;

a(n) + 4*n = A254963(n);

a(n) - n*(n-1)/2 = A147874(n+1);

a(n) + n*(n-1)/2 = A094159(n) (case k=12);

a(n) - n*(n-1) = A062741(n) (see above, this is the case k=9);

a(n) + n*(n-1) = n*(13*n-7)/2 (case k=13);

a(n) - n*(n+1)/2 = A135706(n);

a(n) + n*(n+1)/2 = A033579(n);

a(n) - n*(n+1) = A051682(n);

a(n) + n*(n+1) = A186030(n);

a(n) - n^2 = A062708(n);

a(n) + n^2 = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.

Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011

Sequence found by reading the line from 0, in the direction 0, 8,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A139267 in the same spiral - Omar E. Pol, Sep 09 2011

a(n) is the number of edges of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch May 13 2018

The partial sums of this sequence give A035006. - Leo Tavares, Oct 03 2021

The old definition of this sequence was "Generalized polygonal numbers".

Column T(n,4) of A080853.

Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

A033579 and A033570 interleaved.