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A056105
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First spoke of a hexagonal spiral.
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41
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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
(list;
graph;
refs;
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history;
text;
internal format)
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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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G. C. Greubel, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Grid Graph
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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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).
a(n) = A056106(n) - n = A056107(n) - 2*n.
a(n) = A056108(n) - 3*n = A056109(n) - 4*n = A003215(n) - 5*n.
A008810(3*n-1) = A056109(-n) = a(n). - Michael Somos, Aug 03 2006
G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
From Robert G. Wilson v, Jul 05 2014: (Start)
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
3rd: 330 degrees A056107 3n^2 + 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)
a(n) = 1 + A000567(n). - Omar E. Pol, Apr 26 2017
a(n) = A000290(n-1) + 2*A000290(n), n >= 1. - J. M. Bergot, Mar 03 2018
E.g.f.: (1 + x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018
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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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A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* Harvey P. Dale, Nov 02 2011 *)
Table[3 n^2 - 2 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
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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(PARI) a(n)=3*n^2-2*n+1 /* Michael Somos, Aug 03 2006 */
(Magma) [3*n^2-2*n+1: n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014
(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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Cf. A285792 (prime terms), A113519 (semiprime terms).
Other spokes: A003215, A056106, A056107, A056108, A056109.
Other spirals: A054552.
Cf. A000290, A000567, A008810, A033577.
Sequence in context: A284923 A343850 A235707 * A323891 A212069 A106058
Adjacent sequences: A056102 A056103 A056104 * A056106 A056107 A056108
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley, Jun 09 2000
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STATUS
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approved
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