OFFSET
1,2
COMMENTS
The number of (2 to n) digit sequences that can be found reading in any orientation, including diagonals, in an (n X n) grid. - Paul Cleary, Aug 12 2005
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = (n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3.
From Johannes W. Meijer, Feb 04 2010: (Start)
a(n) = 4 * A162147(n-1). (End)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=12, a(2)=56, a(3)=152. - Harvey P. Dale, Aug 24 2011
From Colin Barker, Mar 11 2012: (Start)
a(n) = 2*n*(1-6*n+5*n^2)/3.
G.f.: 4*x^2*(3+2*x)/(1-x)^4. (End)
E.g.f.: 2*exp(x)*x^2*(9 + 5*x)/3. - Stefano Spezia, Jul 31 2022
EXAMPLE
3 X 3 board: queen has 8*6 moves and 1*8 moves, so a(3)=56.
MATHEMATICA
Table[(n-1)2n^2+(4n^3-6n^2+2n)/3, {n, 40}] (* or *) LinearRecurrence[ {4, -6, 4, -1}, {0, 12, 56, 152}, 40] (* Harvey P. Dale, Aug 24 2011 *)
PROG
(Magma) [(n-1)*2*n^2 + (4*n^3-6*n^2+2*n)/3: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
EXTENSIONS
More terms from Erich Friedman
STATUS
approved
