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%I #96 Dec 31 2022 05:38:19
%S 1,2,9,22,41,66,97,134,177,226,281,342,409,482,561,646,737,834,937,
%T 1046,1161,1282,1409,1542,1681,1826,1977,2134,2297,2466,2641,2822,
%U 3009,3202,3401,3606,3817,4034,4257,4486,4721,4962,5209,5462,5721,5986,6257
%N First spoke of a hexagonal spiral.
%C Also the number of (not necessarily maximal) cliques in the n X n grid graph. - _Eric W. Weisstein_, Nov 29 2017
%H G. C. Greubel, <a href="/A056105/b056105.txt">Table of n, a(n) for n = 0..5000</a>
%H Henry Bottomley, <a href="/A003215/a003215.gif">Illustration of initial terms</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*n^2 - 2*n + 1.
%F a(n) = a(n-1) + 6*n - 5.
%F a(n) = 2*a(n-1) - a(n-2) + 6.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = A056106(n) - n = A056107(n) - 2*n.
%F a(n) = A056108(n) - 3*n = A056109(n) - 4*n = A003215(n) - 5*n.
%F A008810(3*n-1) = A056109(-n) = a(n). - _Michael Somos_, Aug 03 2006
%F G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - _Colin Barker_, Jan 04 2012
%F From _Robert G. Wilson v_, Jul 05 2014: (Start)
%F Each of the 6 primary spokes or rays has a generating formula as stated here:
%F 1st: 90 degrees A056105 3n^2 - 2n + 1
%F 2nd: 30 degrees A056106 3n^2 - n + 1
%F 3rd: 330 degrees A056107 3n^2 + 1
%F 4th: 270 degrees A056108 3n^2 + n + 1
%F 5th: 210 degrees A056109 3n^2 + 2n + 1
%F 6th: 150 degrees A003215 3n^2 + 3n + 1
%F Each of the 6 secondary spokes or rays has a generating formula as stated here:
%F 1st: 60 degrees 12n^2 - 27n + 16
%F 2nd: 360 degrees 12n^2 - 25n + 14
%F 3rd: 300 degrees 12n^2 - 23n + 12
%F 4th: 240 degrees 12n^2 - 21n + 10
%F 5th: 180 degrees 12n^2 - 19n + 8
%F 6th: 120 degrees 12n^2 - 17n + 6 = A033577(n+1)
%F (End)
%F a(n) = 1 + A000567(n). - _Omar E. Pol_, Apr 26 2017
%F a(n) = A000290(n-1) + 2*A000290(n), n >= 1. - _J. M. Bergot_, Mar 03 2018
%F E.g.f.: (1 + x + 3*x^2)*exp(x). - _G. C. Greubel_, Dec 02 2018
%e The spiral begins:
%e 49--48--47--46--45
%e / \
%e 50 28--27--26--25 44
%e / / \ \
%e 51 29 13--12--11 24 43
%e / / / \ \ \
%e 52 30 14 4---3 10 23 42 67
%e / / / / \ \ \ \ \
%e 53 31 15 5 1===2===9==22==41==66==>
%e \ \ \ \ / / / /
%e 54 32 16 6---7---8 21 40 65
%e \ \ \ / / /
%e 55 33 17--18--19--20 39 64
%e \ \ / /
%e 56 34--35--36--37--38 63
%e \ /
%e 57--58--59--60--61--62
%p A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # _Wesley Ivan Hurt_, Jul 06 2014
%t LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* _Harvey P. Dale_, Nov 02 2011 *)
%t Table[3 n^2 - 2 n + 1, {n, 0, 20}] (* _Eric W. Weisstein_, Nov 29 2017 *)
%t CoefficientList[Series[(-1 + x - 6 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)
%o (PARI) a(n)=3*n^2-2*n+1 /* _Michael Somos_, Aug 03 2006 */
%o (Magma) [3*n^2-2*n+1: n in [0..50]]; // _Wesley Ivan Hurt_, Jul 06 2014
%o (Sage) [3*n^2-2*n+1 for n in range(50)] # _G. C. Greubel_, Dec 02 2018
%o (GAP) List([0..50], n -> 3*n^2-2*n+1); # _G. C. Greubel_, Dec 02 2018
%Y Cf. A285792 (prime terms), A113519 (semiprime terms).
%Y Other spokes: A003215, A056106, A056107, A056108, A056109.
%Y Other spirals: A054552.
%Y Cf. A000290, A000567, A008810, A033577.
%K easy,nonn
%O 0,2
%A _Henry Bottomley_, Jun 09 2000
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