OFFSET
0,2
COMMENTS
Also the number of (not necessarily maximal) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017
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).
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).
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
E.g.f.: (1 + x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018
EXAMPLE
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
MAPLE
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 *)
PROG
(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
CROSSREFS
Other spirals: A054552.
KEYWORD
easy,nonn
AUTHOR
Henry Bottomley, Jun 09 2000
STATUS
approved