OFFSET
0,2
COMMENTS
The Granvik array of A172119 is here written in "square": 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 // 1 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 // 1 :: 3 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 // 1 :: 4 :: 7 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 // 1 :: 5 :: 12 :: 15 :: 16 :: 16 :: 16 :: 16 :: 16 :: 16 // 1 :: 6 :: 20 :: 28 :: 31 :: 32 :: 32 :: 32 :: 32 :: 32 // 1 :: 7 :: 33 :: 52 :: 60 :: 63 :: 64 :: 64 :: 64 :: 64 // 1 :: 8 :: 54 :: 96 :: 116 :: 124 :: 127 :: 128 :: 128 :: 128 // 1 :: 9 :: 88 :: 177 :: 224 :: 244 :: 252 :: 255 :: 256 :: 256 //
For n>0 a(n) is also the number of ways to place n^2 non-attacking kings on a 2n X 2n toroidal chessboard. - Vaclav Kotesovec, Aug 28 2011
The number of n-step self-avoiding walks on a 2D square lattice where no step is to a lattice point closer to the origin than the current point. - Scott R. Shannon, Dec 15 2023
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Non-attacking chess pieces, 6th ed, 2013, p. 215.
Index entries for linear recurrences with constant coefficients, signature (3,-2).
FORMULA
G.f.: 4/(1-2*x) - 4/(1-x) + 1.
a(n) = 2^(n+2) - 4 for n>=1, a(0)=1.
a(n) = A028399(n+2), n>0. - R. J. Mathar, Feb 21 2010
a(n) = A240951(n+3) - 4. - Omar E. Pol, Feb 17 2015
a(n) = 2*a(n-1) + 4 for n>1. - J. Conrad, Nov 01 2015
a(n) = 3*a(n-1)-2*a(n-2) for n>2. - Colin Barker, Nov 03 2015
E.g.f.: (1 - 2*exp(x))^2. - Stefano Spezia, May 03 2023
EXAMPLE
a(3) = 2^5 - 4 = 32 - 4 = 28.
MAPLE
taylor(4/(1-2*z)-4/(1-z)+1, z=0, 31);
MATHEMATICA
CoefficientList[Series[4 / (1 - 2 x) - 4 / (1 - x) + 1, {x, 0, 60}], x] (* Vincenzo Librandi, May 29 2013 *)
PROG
(PARI) x='x+O('x^50); Vec(4/(1-2*x)-4/(1-x)+1) \\ Altug Alkan, Nov 02 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Richard Choulet, Feb 07 2010
STATUS
approved