This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A045944 #171 Jul 13 2025 00:40:30
%S A045944 0,5,16,33,56,85,120,161,208,261,320,385,456,533,616,705,800,901,1008,
%T A045944 1121,1240,1365,1496,1633,1776,1925,2080,2241,2408,2581,2760,2945,
%U A045944 3136,3333,3536,3745,3960,4181,4408,4641,4880,5125,5376,5633,5896,6165,6440
%N A045944 Rhombic matchstick numbers: a(n) = n*(3*n+2).
%C A045944 From _Floor van Lamoen_, Jul 21 2001: (Start)
%C A045944 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:
%C A045944 .
%C A045944 85--84--83--82--81--80
%C A045944 . \
%C A045944 56--55--54--53--52 79
%C A045944 / . \ \
%C A045944 57 33--32--31--30 51 78
%C A045944 / / . \ \ \
%C A045944 58 34 16--15--14 29 50 77
%C A045944 / / / . \ \ \ \
%C A045944 59 35 17 5---4 13 28 49 76
%C A045944 / / / / . \ \ \ \ \
%C A045944 60 36 18 6 0 3 12 27 48 75
%C A045944 / / / / / / / / / /
%C A045944 61 37 19 7 1---2 11 26 47 74
%C A045944 \ \ \ \ / / / /
%C A045944 62 38 20 8---9--10 25 46 73
%C A045944 \ \ \ / / /
%C A045944 63 39 21--22--23--24 45 72
%C A045944 \ \ / /
%C A045944 64 40--41--42--43--44 71
%C A045944 \ /
%C A045944 65--66--67--68--69--70
%C A045944 (End)
%C A045944 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
%C A045944 Also, second octagonal numbers. - _Bruno Berselli_, Jan 13 2011
%C A045944 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
%C A045944 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
%C A045944 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
%C A045944 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
%H A045944 Ivan Panchenko, <a href="/A045944/b045944.txt">Table of n, a(n) for n = 0..1000</a>
%H A045944 Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H A045944 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H A045944 Leo Tavares, <a href="/A045944/a045944_1.jpg">Illustration: Square Stars</a>
%H A045944 Leo Tavares, <a href="/A045944/a045944_2.jpg">Illustration: Split Stars</a>
%H A045944 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A045944 O.g.f.: x*(5+x)/(1-x)^3. - _R. J. Mathar_, Jan 07 2008
%F A045944 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
%F A045944 a(n) = a(n-1) + 6*n - 1 (with a(0)=0). - _Vincenzo Librandi_, Nov 18 2010
%F A045944 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
%F A045944 a(n) = Sum_{i=0..n-1} A016969(i) for n > 0. - _Bruno Berselli_, Jan 13 2011
%F A045944 a(n) = A174709(6*n+4). - _Philippe Deléham_, Mar 26 2013
%F A045944 a(n) = A001082(2*n). - _Michael Turniansky_, Aug 24 2013
%F A045944 Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - _Vaclav Kotesovec_, Apr 27 2016
%F A045944 a(n) = A002378(n) + A014105(n). - _J. M. Bergot_, Apr 24 2018
%F A045944 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - _Amiram Eldar_, Jul 03 2020
%F A045944 E.g.f.: exp(x)*x*(5 + 3*x). - _Stefano Spezia_, Jun 08 2021
%F A045944 From _Leo Tavares_, Oct 14 2021: (Start)
%F A045944 a(n) = A000290(n) + 4*A000217(n). See Square Stars illustration.
%F A045944 a(n) = A000567(n+2) - A022144(n+1)
%F A045944 a(n) = A005563(n) + A001105(n).
%F A045944 a(n) = A056109(n) - 1. (End)
%F A045944 From _Leo Tavares_, Oct 06 2022: (Start)
%F A045944 a(n) = A003154(n+1) - A000567(n+1). See Split Stars illustration.
%F A045944 a(n) = A014105(n) + 2*A000217(n). (End)
%t A045944 Table[n*(3n+2), {n,0,60}] (* _Harvey P. Dale_, May 05 2011 *)
%t A045944 LinearRecurrence[{3,-3,1},{0,5,16},60] (* _Harvey P. Dale_, Jan 19 2016 *)
%t A045944 CoefficientList[Series[x*(5 + x)/(1 - x)^3,{x, 0, 60}], x] (* _Stefano Spezia_, Sep 01 2018 *)
%o A045944 (PARI) a(n)=n*(3*n+2) \\ _Charles R Greathouse IV_, Nov 20 2012
%o A045944 (Magma) [n*(3*n+2) : n in [0..100]]; // _Wesley Ivan Hurt_, Sep 24 2017
%Y A045944 Bisection of A001859. See Comments of A135713.
%Y A045944 Cf. A000217, A000567, A001082, A002378, A016969, A049450, A174709.
%Y A045944 Cf. second n-gonal numbers: A005449, A014105, A147875, A179986, A033954, A062728, A135705.
%Y A045944 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.
%Y A045944 Cf. A000290, A022144, A005563, A001105.
%Y A045944 Cf. A056109.
%Y A045944 Cf. A003154.
%K A045944 nonn,easy,nice
%O A045944 0,2
%A A045944 _R. K. Guy_