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A126474 Number of arrays in [1..6]^n with adjacent elements differing by three or less. 2
1, 6, 30, 154, 788, 4034, 20650, 105708, 541122, 2770018, 14179796, 72586754, 371573530, 1902094812, 9736874082, 49843318162, 255149275268, 1306115946338, 6686042370634, 34226029248972, 175203956722818 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

[Empirical] a(base,n)=a(base-1,n)+7^(n-1) for base>=3n-2; a(base,n)=a(base-1,n)+7^(n-1)-2 when base=3n-3

Original name: Number of base 6 n-digit numbers with adjacent digits differing by three or less.

LINKS

Robert Israel, Table of n, a(n) for n = 0..1408

FORMULA

Conjectures from Colin Barker, Jan 20 2017: (Start)

a(n) = 5*a(n-1) + a(n-2) - 2*a(n-3) for n>2.

G.f.: (1 + x - x^2) / (1 - 5*x - x^2 + 2*x^3).

(End)

From Robert Israel, Jan 23 2018: (Start)

a(n) = e^T M^(n-1) e where e = [1,1,1,1,1,1]^T and M is the 6 X 6 matrix with entries M(i,j) = 1 if |i-j|<=3, 0 otherwise.

The fact that (M^3-5*M^2-M+2I) e = 0 implies Colin Barker's recursion, and the G.f. follows. (End)

EXAMPLE

For n=2 the a(2)=30 solutions are [1, 1], [1, 2], [1, 3], [1, 4], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [4, 6], [5, 2], [5, 3], [5, 4], [5, 5], [5, 6], [6, 3], [6, 4], [6, 5], [6, 6]. - Robert Israel, Jan 23 2018

MAPLE

f:= gfun:-rectoproc({a(n) = 5*a(n-1) + a(n-2) - 2*a(n-3), a(0)=1, a(1)=6, a(2)=30}, a(n), remember):

map(f, [$0..30]); # Robert Israel, Jan 23 2018

PROG

(S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>3)+($[i+1]`-$[i]`>3))

CROSSREFS

Cf. Base 6 differing by two or less A126393, one or less A126360.

Sequence in context: A033132 A022023 A066534 * A127017 A152223 A152224

Adjacent sequences:  A126471 A126472 A126473 * A126475 A126476 A126477

KEYWORD

nonn,base

AUTHOR

R. H. Hardin, Dec 27 2006

STATUS

approved

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Last modified June 12 08:13 EDT 2021. Contains 344943 sequences. (Running on oeis4.)