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A056105
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First spoke of a hexagonal spiral.
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45
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1, 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, 342, 409, 482, 561, 646, 737, 834, 937, 1046, 1161, 1282, 1409, 1542, 1681, 1826, 1977, 2134, 2297, 2466, 2641, 2822, 3009, 3202, 3401, 3606, 3817, 4034, 4257, 4486, 4721, 4962, 5209, 5462, 5721, 5986, 6257
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OFFSET
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0,2
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COMMENTS
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Also the number of (not necessarily maximal) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017
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LINKS
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Eric Weisstein's World of Mathematics, Clique
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FORMULA
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a(n) = 3*n^2 - 2*n + 1.
a(n) = a(n-1) + 6*n - 5.
a(n) = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
Each of the 6 primary spokes or rays has a generating formula as stated here:
1st: 90 degrees A056105 3n^2 - 2n + 1
2nd: 30 degrees A056106 3n^2 - n + 1
4th: 270 degrees A056108 3n^2 + n + 1
5th: 210 degrees A056109 3n^2 + 2n + 1
6th: 150 degrees A003215 3n^2 + 3n + 1
Each of the 6 secondary spokes or rays has a generating formula as stated here:
1st: 60 degrees 12n^2 - 27n + 16
2nd: 360 degrees 12n^2 - 25n + 14
3rd: 300 degrees 12n^2 - 23n + 12
4th: 240 degrees 12n^2 - 21n + 10
5th: 180 degrees 12n^2 - 19n + 8
6th: 120 degrees 12n^2 - 17n + 6 = A033577(n+1)
(End)
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EXAMPLE
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The spiral begins:
49--48--47--46--45
/ \
50 28--27--26--25 44
/ / \ \
51 29 13--12--11 24 43
/ / / \ \ \
52 30 14 4---3 10 23 42 67
/ / / / \ \ \ \ \
53 31 15 5 1===2===9==22==41==66==>
\ \ \ \ / / / /
54 32 16 6---7---8 21 40 65
\ \ \ / / /
55 33 17--18--19--20 39 64
\ \ / /
56 34--35--36--37--38 63
\ /
57--58--59--60--61--62
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* Harvey P. Dale, Nov 02 2011 *)
CoefficientList[Series[(-1 + x - 6 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)
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PROG
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(Sage) [3*n^2-2*n+1 for n in range(50)] # G. C. Greubel, Dec 02 2018
(GAP) List([0..50], n -> 3*n^2-2*n+1); # G. C. Greubel, Dec 02 2018
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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