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%I #161 Sep 04 2023 17:24:12
%S 0,5,16,33,56,85,120,161,208,261,320,385,456,533,616,705,800,901,1008,
%T 1121,1240,1365,1496,1633,1776,1925,2080,2241,2408,2581,2760,2945,
%U 3136,3333,3536,3745,3960,4181,4408,4641,4880,5125,5376,5633,5896,6165,6440
%N Rhombic matchstick numbers: a(n) = n*(3*n+2).
%C From _Floor van Lamoen_, Jul 21 2001: (Start)
%C 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 .
%C 85--84--83--82--81--80
%C . \
%C 56--55--54--53--52 79
%C / . \ \
%C 57 33--32--31--30 51 78
%C / / . \ \ \
%C 58 34 16--15--14 29 50 77
%C / / / . \ \ \ \
%C 59 35 17 5---4 13 28 49 76
%C / / / / . \ \ \ \ \
%C 60 36 18 6 0 3 12 27 48 75
%C / / / / / / / / / /
%C 61 37 19 7 1---2 11 26 47 74
%C \ \ \ \ / / / /
%C 62 38 20 8---9--10 25 46 73
%C \ \ \ / / /
%C 63 39 21--22--23--24 45 72
%C \ \ / /
%C 64 40--41--42--43--44 71
%C \ /
%C 65--66--67--68--69--70
%C (End)
%C 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 Also, second octagonal numbers. - _Bruno Berselli_, Jan 13 2011
%C 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 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 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 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 Ivan Panchenko, <a href="/A045944/b045944.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H Leo Tavares, <a href="/A045944/a045944_1.jpg">Illustration: Square Stars</a>
%H Leo Tavares, <a href="/A045944/a045944_2.jpg">Illustration: Split Stars</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F O.g.f.: x*(5+x)/(1-x)^3. - _R. J. Mathar_, Jan 07 2008
%F 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 a(n) = a(n-1) + 6*n - 1 (with a(0)=0). - _Vincenzo Librandi_, Nov 18 2010
%F 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 a(n) = Sum_{i=0..n-1} A016969(i) for n > 0. - _Bruno Berselli_, Jan 13 2011
%F a(n) = A174709(6*n+4). - _Philippe Deléham_, Mar 26 2013
%F a(n) = A001082(2*n). - _Michael Turniansky_, Aug 24 2013
%F Sum_{n>=1} 1/a(n) = (9 + sqrt(3)*Pi - 9*log(3))/12 = 0.3794906245574721941... . - _Vaclav Kotesovec_, Apr 27 2016
%F a(n) = A002378(n) + A014105(n). - _J. M. Bergot_, Apr 24 2018
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(12) - 3/4. - _Amiram Eldar_, Jul 03 2020
%F E.g.f.: exp(x)*x*(5 + 3*x). - _Stefano Spezia_, Jun 08 2021
%F From _Leo Tavares_, Oct 14 2021: (Start)
%F a(n) = A000290(n) + 4*A000217(n). See Square Stars illustration.
%F a(n) = A000567(n+2) - A022144(n+1)
%F a(n) = A005563(n) + A001105(n).
%F a(n) = A056109(n) - 1. (End)
%F From _Leo Tavares_, Oct 06 2022: (Start)
%F a(n) = A003154(n+1) - A000567(n+1). See Split Stars illustration.
%F a(n) = A014105(n) + 2*A000217(n). (End)
%t Table[n*(3n+2), {n,0,60}] (* _Harvey P. Dale_, May 05 2011 *)
%t LinearRecurrence[{3,-3,1},{0,5,16},60] (* _Harvey P. Dale_, Jan 19 2016 *)
%t CoefficientList[Series[x*(5 + x)/(1 - x)^3,{x, 0, 60}], x] (* _Stefano Spezia_, Sep 01 2018 *)
%o (PARI) a(n)=n*(3*n+2) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [n*(3*n+2) : n in [0..100]]; // _Wesley Ivan Hurt_, Sep 24 2017
%Y Bisection of A001859. See Comments of A135713.
%Y Cf. A000217, A000567, A001082, A002378, A016969, A049450, A174709.
%Y Cf. second n-gonal numbers: A005449, A014105, A147875, A179986, A033954, A062728, A135705.
%Y 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 Cf. A000290, A022144, A005563, A001105.
%Y Cf. A056109.
%Y Cf. A003154.
%K nonn,easy,nice
%O 0,2
%A _R. K. Guy_
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