A206454 Number of 0..7 arrays of length n avoiding the consecutive pattern 0..7.

8, 64, 512, 4096, 32768, 262144, 2097152, 16777215, 134217712, 1073741632, 8589932544, 68719456256, 549755617280, 4398044676096, 35184355311616, 281474825715713, 2251798471507992, 18014386698322304, 144115084996645888, 1152920611253710848, 9223364340274069504

Offset: 1

Author

R. H. Hardin, Feb 07 2012

Keywords

• nonn
• easy

Comments

Column 7 of A206455.

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Index entries for linear recurrences with constant coefficients, signature (8,0,0,0,0,0,0,-1).

Crossrefs

Cf. A206455.

Programs

Mathematica

Table[Sum[(-1)^i*8^(n-8*i)*Binomial[n-7*i,i],{i,0,Floor[n/8]}],{n,20}] (* Stefano Spezia, Feb 12 2023 *)
LinearRecurrence[{8,0,0,0,0,0,0,-1},{8,64,512,4096,32768,262144,2097152,16777215},30] (* Harvey P. Dale, Jan 10 2025 *)

Formula

a(n) = 8*a(n-1) - a(n-8).

a(n) = Sum_{i=0..floor(n/8)} (-1)^i*8^(n-8*i)*binomial(n-7*i,i).

G.f.: x*(8 - x^7) / (1 - 8*x + x^8). - Colin Barker, Jun 16 2018