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 A045944 Rhombic matchstick numbers: a(n) = n*(3*n+2). 71
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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). 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) 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 *) PROG (PARI) a(n)=n*(3*n+2) \\ Charles R Greathouse IV, Nov 20 2012 (Magma) [n*(3*n+2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 24 2017 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. Sequence in context: A063076 A270805 A132479 * A038361 A227719 A172166 Adjacent sequences: A045941 A045942 A045943 * A045945 A045946 A045947 KEYWORD nonn,easy,nice AUTHOR R. K. Guy STATUS approved

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Last modified November 30 01:30 EST 2023. Contains 367452 sequences. (Running on oeis4.)