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%I #112 Oct 14 2022 02:21:00
%S 0,2,10,24,44,70,102,140,184,234,290,352,420,494,574,660,752,850,954,
%T 1064,1180,1302,1430,1564,1704,1850,2002,2160,2324,2494,2670,2852,
%U 3040,3234,3434,3640,3852,4070,4294,4524,4760,5002,5250,5504,5764
%N Pentagonal numbers multiplied by 2: a(n) = n*(3*n-1).
%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 sequence found by reading the line from 0 in the direction 0,2,.... The spiral begins:
%C .
%C 56--55--54--53--52
%C / \
%C 57 33--32--31--30 51
%C / / \ \
%C 58 34 16--15--14 29 50
%C / / / \ \ \
%C 59 35 17 5---4 13 28 49
%C / / / / \ \ \ \
%C 60 36 18 6 0 3 12 27 48
%C / / / / / . / / / /
%C 61 37 19 7 1---2 11 26 47
%C \ \ \ \ . / / /
%C 62 38 20 8---9--10 25 46
%C \ \ \ . / /
%C 63 39 21--22--23--24 45
%C \ \ . /
%C 64 40--41--42--43--44
%C \ .
%C 65--66--67--68--69--70
%C (End)
%C Starting with offset 1 = binomial transform of [2, 8, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, Jan 09 2009
%C Number of possible pawn moves on an (n+1) X (n+1) chessboard (n=>3). - _Johannes W. Meijer_, Feb 04 2010
%C a(n) = A069905(6n-1): Number of partitions of 6*n-1 into 3 parts. - _Adi Dani_, Jun 04 2011
%C Even octagonal numbers divided by 4. - _Omar E. Pol_, Aug 19 2011
%C Partial sums give A011379. - _Omar E. Pol_, Jan 12 2013
%C First differences are A016933; second differences equal 6. - _Bob Selcoe_, Apr 02 2015
%C For n >= 1, the continued fraction expansion of sqrt(27*a(n)) is [9n-2; {2, 2n-1, 6, 2n-1, 2, 18n-4}]. - _Magus K. Chu_, Oct 13 2022
%H Ivan Panchenko, <a href="/A049450/b049450.txt">Table of n, a(n) for n = 0..1000</a>
%H Richard P. Brent, <a href="http://arxiv.org/abs/1407.3533">Generalising Tuenter's binomial sums</a>, arXiv:1407.3533 [math.CO], 2014. (page 16)
%H Leo Tavares, <a href="/A049450/a049450.jpg">Illustration: X Hexagons</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.: A(x) = 2*x*(1+2*x)/(1-x)^3.
%F a(n) = A049452(n) - A033428(n). - _Zerinvary Lajos_, Jun 12 2007
%F a(n) = 2*A000326(n), twice pentagonal numbers. - _Omar E. Pol_, May 14 2008
%F a(n) = A022264(n) - A000217(n). - _Reinhard Zumkeller_, Oct 09 2008
%F a(n) = a(n-1) + 6*n - 4 (with a(0)=0). - _Vincenzo Librandi_, Aug 06 2010
%F a(n) = A014642(n)/4 = A033579(n)/2. - Omar E. Pol, Aug 19 2011
%F a(n) = A000290(n) + A000384(n) = A000217(n) + A000566(n). - _Omar E. Pol_, Jan 11 2013
%F a(n+1) = A014107(n+2) + A000290(n). - _Philippe Deléham_, Mar 30 2013
%F E.g.f.: x*(2 + 3*x)*exp(x). - _Vincenzo Librandi_, Apr 28 2016
%F a(n) = (2/3)*A000217(3*n-1). - _Bruno Berselli_, Feb 13 2017
%F a(n) = A002061(n) + A056220(n). - _Bruce J. Nicholson_, Sep 21 2017
%F From _Amiram Eldar_, Feb 20 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 3*log(3)/2 - Pi/(2*sqrt(3)).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(3) - 2*log(2). (End)
%F From _Leo Tavares_, Feb 23 2022: (Start)
%F a(n) = A003215(n) - A016813(n).
%F a(n) = 2*A000290(n) + 2*A000217(n-1). (End)
%e On a 4 X 4 chessboard pawns at the second row have (3+4+4+3) moves and pawns at the third row have (2+3+3+2) moves so a(3) = 24. - _Johannes W. Meijer_, Feb 04 2010
%e From _Adi Dani_, Jun 04 2011: (Start)
%e a(1)=2: the partitions of 6*1-1=5 into 3 parts are [1,1,3] and[1,2,2].
%e a(2)=10: the partitions of 6*2-1=11 into 3 parts are [1,1,9], [1,2,8], [1,3,7], [1,4,6], [1,5,5], [2,2,7], [2,3,6], [2,4,5], [3,3,5], and [3,4,4].
%e (End)
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%e . 2 10 24 44 70
%e - _Philippe Deléham_, Mar 30 2013
%p seq(n*(3*n-1),n=0..44); # _Zerinvary Lajos_, Jun 12 2007
%t Table[n(3n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,2,10},50] (* _Harvey P. Dale_, Jun 21 2014 *)
%t 2*PolygonalNumber[5,Range[0,50]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 01 2018 *)
%o (PARI) a(n)=n*(3*n-1) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [n*(3*n-1) : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 24 2017
%o (Sage) [n*(3*n-1) for n in (0..50)] # _G. C. Greubel_, Aug 31 2019
%o (GAP) List([0..50], n-> n*(3*n-1)); # _G. C. Greubel_, Aug 31 2019
%Y Cf. A000567.
%Y Bisection of A001859. Cf. A045944, A000326, A033579, A027599, A049451.
%Y Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A002492 (Bishop).
%Y Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488. [_Bruno Berselli_, Jun 10 2013]
%Y Cf. sequences listed in A254963.
%Y Cf. A003215, A016813, A000290, A000217.
%K nonn,easy,nice
%O 0,2
%A Joe Keane (jgk(AT)jgk.org).
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