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A126360 Number of base 6 n-digit numbers with adjacent digits differing by one or less. 7
1, 6, 16, 44, 122, 340, 950, 2658, 7442, 20844, 58392, 163594, 458356, 1284250, 3598338, 10082246, 28249720, 79153804, 221783810, 621424108, 1741191198, 4878708658, 13669836930, 38302030548, 107319902744, 300703682402 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Leading 0's are allowed. - Robert Israel, Aug 12 2019

LINKS

Robert Israel, Table of n, a(n) for n = 0..2231

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

FORMULA

Conjecture: a(n) = 4*a(n-1)-3*a(n-2)-a(n-3) for n>3. G.f.: -(x^3+5*x^2-2*x-1)/(x^3+3*x^2-4*x+1). [Colin Barker, Nov 26 2012]

From Robert Israel, Aug 12 2019: (Start)

a(n) = e^T A^(n-1) e for n>=1, where A is the 6 X 6 matrix with 1 on the main diagonal and first super- and sub-diagonals, 0 elsewhere, and e the column vector (1,1,1,1,1,1).

Barker's conjecture follows from the fact that (A^3-4*A^2+3*A+1) e = 0. (End)

MAPLE

A:=LinearAlgebra:-ToeplitzMatrix([1, 1, 0, 0, 0, 0], symmetric):

e:= Vector(6, 1):

1, seq(e^%T . A^n . e, n=0..30); # Robert Israel, Aug 12 2019

PROG

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

CROSSREFS

Sequence in context: A317758 A010915 A260384 * A264545 A296855 A105465

Adjacent sequences:  A126357 A126358 A126359 * A126361 A126362 A126363

KEYWORD

nonn,base

AUTHOR

R. H. Hardin, Dec 26 2006

STATUS

approved

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Last modified February 16 15:17 EST 2020. Contains 331961 sequences. (Running on oeis4.)