A028896 6 times triangular numbers: a(n) = 3*n*(n+1).

0, 6, 18, 36, 60, 90, 126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, 1260, 1386, 1518, 1656, 1800, 1950, 2106, 2268, 2436, 2610, 2790, 2976, 3168, 3366, 3570, 3780, 3996, 4218, 4446, 4680, 4920, 5166, 5418, 5676

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

Keywords

• nonn
• easy

Comments

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, 6, ...

The spiral begins:

85--84--83--82--81--80

/ \

86 56--55--54--53--52 79

/ / \ \

87 57 33--32--31--30 51 78

/ / / \ \ \

88 58 34 16--15--14 29 50 77

/ / / / \ \ \ \

89 59 35 17 5---4 13 28 49 76

/ / / / / \ \ \ \ \

<==90==60==36==18===6===0 3 12 27 48 75

/ / / / / / / / / /

61 37 19 7 1---2 11 26 47 74

\ \ \ \ / / / /

62 38 20 8---9--10 25 46 73

\ \ \ / / /

63 39 21--22--23--24 45 72

\ \ / /

64 40--41--42--43--44 71

\ /

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

(End)

If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007

a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008

Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011

Partial sums of A008588. - R. J. Mathar, Aug 28 2014

Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017

a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices. The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices). (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
• Leo Tavares, Illustration: Centroid Hexagons.
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080.

Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles).

Cf. A007531.

The partial sums give A007531. - Leo Tavares, Jan 22 2022

Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

Programs

GAP

List([0..44],n->3*n*(n+1)); # Muniru A Asiru, Mar 15 2019

Magma

[3*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

Maple

[seq(6*binomial(n,2),n=1..44)]; # Zerinvary Lajos, Nov 24 2006

Mathematica

6 Accumulate[Range[0, 50]] (* Harvey P. Dale, Mar 05 2012 *)
6 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* Eric W. Weisstein, Jul 27 2017 *)

PARI

a(n)=3*n*(n+1) \\ Charles R Greathouse IV, Sep 24 2015

PARI

first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ Iain Fox, Feb 14 2018

Python

def A028896(n): return 3*n*(n+1) # Chai Wah Wu, Aug 07 2025

Formula

O.g.f.: 6*x/(1 - x)^3.

E.g.f.: 3*x*(x + 2)*exp(x). - G. C. Greubel, Aug 19 2017

a(n) = 6*A000217(n).

a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003

From Zerinvary Lajos, Mar 06 2007: (Start)

a(n) = A049598(n)/2.

a(n) = A124080(n) - A046092(n).

a(n) = A033996(n) - A002378(n). (End)

a(n) = A002378(n)*3 = A045943(n)*2. - Omar E. Pol, Dec 12 2008

a(n) = a(n-1) + 6*n for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010

a(n) = A003215(n) - 1. - Omar E. Pol, Oct 03 2011

From Philippe Deléham, Mar 26 2013: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18.

a(n) = A174709(6*n + 5). (End)

a(n) = A049450(n) + 4*n. - Lear Young, Apr 24 2014

a(n) = Sum_{i = n..2*n} 2*i. - Bruno Berselli, Feb 14 2018

a(n) = A320047(1, n, 1). - Kolosov Petro, Oct 04 2018

a(n) = T(3*n) - T(2*n-2) + T(n-2), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 04 2020

From Amiram Eldar, Feb 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/3.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 1/3. (End)

From Amiram Eldar, Feb 21 2023: (Start)

Product_{n>=1} (1 - 1/a(n)) = -(3/Pi)*cos(sqrt(7/3)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (3/Pi)*cosh(Pi/(2*sqrt(3))). (End)