A336627 Coordination sequence for the Manhattan lattice.

1, 2, 4, 8, 11, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224

Offset

0,2

Comments

In the Manhattan lattice, N-S streets run alternately N and S, and E-W streets run alternately E and W. - N. J. A. Sloane, Jul 29 2020

Links

Table of n, a(n) for n=0..57.

Sean A. Irvine, Illustration of a(0) to a(7)

N. J. A. Sloane, Crude drawing of initial layers showing paths of length 6 from origin (looking North-West). The presence of three points at distance 4 from the origin on the line of symmetry explains why a(4) is odd!

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

G.f.: (1+x^2) * (1+2*x^3-x^4) / (1-x)^2.

a(n) = 4*(n-1), n >= 5.

Mathematica

CoefficientList[Series[(1+x^2)(1+2x^3-x^4)/(1-x)^2, {x, 0, 80}], x] (* or *) LinearRecurrence[{2, -1}, {1, 2, 4, 8, 11, 16, 20}, 80] (* Harvey P. Dale, Dec 28 2021 *)

Prog

(PARI) a(n)=if(n>4, 4*n-4, min(2^n, 11)) \\ Charles R Greathouse IV, Oct 18 2022

Crossrefs

Cf. A008574 (square lattice), A117633 (self-avoiding walks).

Sequence in context: A326299 A320448 A115797 * A242313 A003596 A018600

Adjacent sequences: A336624 A336625 A336626 * A336628 A336629 A336630

Keyword

nonn,nice,easy

Author

Sean A. Irvine, Jul 28 2020

Status

approved