A173033 Second diagonal under the main diagonal in A172119 written in a square (see comment).

1, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068, 262140, 524284, 1048572, 2097148, 4194300, 8388604, 16777212, 33554428, 67108860, 134217724, 268435452, 536870908, 1073741820, 2147483644, 4294967292, 8589934588, 17179869180

Offset: 0

Author

Richard Choulet, Feb 07 2010

Keywords

• easy
• nonn

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

Examples

			a(3) = 2^5 - 4 = 32 - 4 = 28.
		

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).

Crossrefs

Cf. A000079, A028399, A172119, A240951.

Programs

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 *)

PARI

my(x='x+O('x^50)); Vec(4/(1-2*x)-4/(1-x)+1) \\ Altug Alkan, Nov 02 2015

Sage

def a(n): return 1 if 0==n else 2**(n+2) - 4 # Torlach Rush, Jan 09 2025

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