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A126362
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Number of base 8 n-digit numbers with adjacent digits differing by one or less.
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8
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1, 8, 22, 62, 176, 502, 1436, 4116, 11814, 33942, 97582, 280676, 807574, 2324116, 6689624, 19257202, 55439298, 159611886, 459545688, 1323132230, 3809653732, 10969153364, 31583803574, 90940708414, 261850874726, 753964626300
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OFFSET
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0,2
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COMMENTS
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[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.
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LINKS
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FORMULA
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Conjecture: a(n) = 5*a(n-1) - 6*a(n-2) - a(n-3) + 2*a(n-4) for n > 4.
G.f.: -(4*x^4 + x^3 - 12*x^2 + 3*x + 1)/((2*x - 1)*(x^3 - 3*x + 1)). (End)
a(n) = e^T A^(n-1) e for n>=1, where A is the 8 X 8 matrix with 1 on the main diagonal and first super- and subdiagonals, 0 elsewhere, and e the column vector (1,1,1,1,1,1,1,1). Barker's conjecture follows from the fact that (A^4 - 5*A^3 + 6*A^2 + A - 2*I)*e = 0. (End)
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MAPLE
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f:= gfun:-rectoproc({a(n)=5*a(n-1)-6*a(n-2)-a(n-3)+2*a(n-4), a(0)=1, a(1)=8, a(2)=22, a(3)=62, a(4)=176}, a(n), remember):
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PROG
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(S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>1)+($[i+1]`-$[i]`>1))
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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