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A049450
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Pentagonal numbers multiplied by 2: a(n) = n*(3*n-1).
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48
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0, 2, 10, 24, 44, 70, 102, 140, 184, 234, 290, 352, 420, 494, 574, 660, 752, 850, 954, 1064, 1180, 1302, 1430, 1564, 1704, 1850, 2002, 2160, 2324, 2494, 2670, 2852, 3040, 3234, 3434, 3640, 3852, 4070, 4294, 4524, 4760, 5002, 5250, 5504, 5764
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OFFSET
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0,2
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COMMENTS
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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:
.
56--55--54--53--52
/ \
57 33--32--31--30 51
/ / \ \
58 34 16--15--14 29 50
/ / / \ \ \
59 35 17 5---4 13 28 49
/ / / / \ \ \ \
60 36 18 6 0 3 12 27 48
/ / / / / . / / / /
61 37 19 7 1---2 11 26 47
\ \ \ \ . / / /
62 38 20 8---9--10 25 46
\ \ \ . / /
63 39 21--22--23--24 45
\ \ . /
64 40--41--42--43--44
\ .
65--66--67--68--69--70
(End)
Starting with offset 1 = binomial transform of [2, 8, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jan 09 2009
Number of possible pawn moves on an (n+1) X (n+1) chessboard (n=>3). - Johannes W. Meijer, Feb 04 2010
a(n) = A069905(6n-1): Number of partitions of 6*n-1 into 3 parts. - Adi Dani, Jun 04 2011
Even octagonal numbers divided by 4. - Omar E. Pol, Aug 19 2011
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
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LINKS
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FORMULA
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O.g.f.: A(x) = 2*x*(1+2*x)/(1-x)^3.
Sum_{n>=1} 1/a(n) = 3*log(3)/2 - Pi/(2*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/sqrt(3) - 2*log(2). (End)
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EXAMPLE
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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
a(1)=2: the partitions of 6*1-1=5 into 3 parts are [1,1,3] and[1,2,2].
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].
(End)
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. 2 10 24 44 70
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MAPLE
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MATHEMATICA
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Table[n(3n-1), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 2, 10}, 50] (* Harvey P. Dale, Jun 21 2014 *)
2*PolygonalNumber[5, Range[0, 50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 01 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Joe Keane (jgk(AT)jgk.org).
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STATUS
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approved
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