A056105 First spoke of a hexagonal spiral.

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

Offset: 0

Author

Henry Bottomley, Jun 09 2000

Keywords

• easy
• nonn

Comments

Also the number of (not necessarily maximal) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017

Examples

			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
		

Links

• 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).

Crossrefs

Cf. A285792 (prime terms), A113519 (semiprime terms).

Other spokes: A003215, A056106, A056107, A056108, A056109.

Other spirals: A054552.

Cf. A000290, A000567, A008810, A033577.

Programs

GAP

List([0..50], n -> 3*n^2-2*n+1); # G. C. Greubel, Dec 02 2018

Magma

[3*n^2-2*n+1: n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014

Maple

A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014

Mathematica

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 *)

PARI

a(n)=3*n^2-2*n+1 /* Michael Somos, Aug 03 2006 */

Sage

[3*n^2-2*n+1 for n in range(50)] # G. C. Greubel, Dec 02 2018

Formula

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