A045944 Rhombic matchstick numbers: a(n) = n*(3*n+2).

0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008, 1121, 1240, 1365, 1496, 1633, 1776, 1925, 2080, 2241, 2408, 2581, 2760, 2945, 3136, 3333, 3536, 3745, 3960, 4181, 4408, 4641, 4880, 5125, 5376, 5633, 5896, 6165, 6440

Offset: 0

Author

R. K. Guy

Keywords

• nonn
• easy
• nice

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 n-th term of the sequence found by reading the line from 0 in the direction 0,5,.... The spiral begins:

.

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

. \

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

/ . \ \

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

/ / . \ \ \

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

/ / / . \ \ \ \

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

/ / / / . \ \ \ \ \

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)

Connection to triangular numbers: a(n) = 4*T_n + S_n where T_n is the n-th triangular number and S_n is the n-th square. - William A. Tedeschi, Sep 12 2010

Also, second octagonal numbers. - Bruno Berselli, Jan 13 2011

Sequence found by reading the line from 0, in the direction 0, 16, ... and the line from 5, in the direction 5, 33, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

Let P denote the points from the n X n grid. A(n-1) also coincides with the minimum number of points Q needed to "block" P, that is, every line segment spanned by two points from P must contain one point from Q. - Manfred Scheucher, Aug 30 2018

Also the number of internal edges of an (n+1)*(n+1) "square" of hexagons; i.e., n+1 rows, each of n+1 edge-adjacent hexagons, stacked with minimal overhang. - Jon Hart, Sep 29 2019

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

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
• M. Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
• Leo Tavares, Illustration: Square Stars
• Leo Tavares, Illustration: Split Stars
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A001859. See Comments of A135713.

Cf. A000217, A000567, A001082, A002378, A016969, A049450, A174709.

Cf. second n-gonal numbers: A005449, A014105, A147875, A179986, A033954, A062728, A135705.

Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734, A139273.

Cf. A000290, A022144, A005563, A001105.

Cf. A056109.

Cf. A003154.

Programs

Magma

[n*(3*n+2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 24 2017

Mathematica

Table[n*(3n+2), {n,0,60}] (* Harvey P. Dale, May 05 2011 *)
LinearRecurrence[{3,-3,1},{0,5,16},60] (* Harvey P. Dale, Jan 19 2016 *)
CoefficientList[Series[x*(5 + x)/(1 - x)^3,{x, 0, 60}], x] (* Stefano Spezia, Sep 01 2018 *)

PARI

a(n)=n*(3*n+2) \\ Charles R Greathouse IV, Nov 20 2012

Formula

O.g.f.: x*(5+x)/(1-x)^3. - R. J. Mathar, Jan 07 2008

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=5, a(2)=16. - Harvey P. Dale, May 06 2011

a(n) = a(n-1) + 6*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010

For n > 0, a(n)^3 + (a(n)+1)^3 + ... + (a(n)+n)^3 + 2*A000217(n)^2 = (a(n) + n + 1)^3 + ... + (a(n) + 2n)^3; see also A033954. - Charlie Marion, Dec 08 2007

a(n) = Sum_{i=0..n-1} A016969(i) for n > 0. - Bruno Berselli, Jan 13 2011

a(n) = A174709(6*n+4). - Philippe Deléham, Mar 26 2013

a(n) = A001082(2*n). - Michael Turniansky, Aug 24 2013

Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - Vaclav Kotesovec, Apr 27 2016

a(n) = A002378(n) + A014105(n). - J. M. Bergot, Apr 24 2018

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - Amiram Eldar, Jul 03 2020

E.g.f.: exp(x)*x*(5 + 3*x). - Stefano Spezia, Jun 08 2021

From Leo Tavares, Oct 14 2021: (Start)

a(n) = A000290(n) + 4*A000217(n). See Square Stars illustration.

a(n) = A000567(n+2) - A022144(n+1)

a(n) = A005563(n) + A001105(n).

a(n) = A056109(n) - 1. (End)

From Leo Tavares, Oct 06 2022: (Start)

a(n) = A003154(n+1) - A000567(n+1). See Split Stars illustration.

a(n) = A014105(n) + 2*A000217(n). (End)