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A126501
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Number of n-tuples of numbers [0..5] (leading zeros allowed) in which adjacent digits differ by 4 or less.
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16
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1, 6, 34, 194, 1106, 6306, 35954, 204994, 1168786, 6663906, 37994674, 216628994, 1235123666, 7042134306, 40151166194, 228924368194, 1305226505746, 7441830001506, 42430056030514, 241917600158594, 1379308224915026
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OFFSET
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0,2
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COMMENTS
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For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,5} containing no subwords 00 and 11. - Milan Janjic, Jan 31 2015
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LINKS
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FORMULA
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[Empirical] a(base,n) = a(base-1,n)+9^(n-1) for base>=4n-3; a(base,n) = a(base-1,n)+9^(n-1)-2 when base=4n-4.
G.f.: (1+x)/(1-5*x-4*x^2).
a(n) = 5*a(n-1) + 4*a(n-2), a(0) = 1, a(1) = 6.
a(n) = Sum_{k, 0<=k<=n} A054458(n,k)*3^k. (End)
Conjecture: a(n) = (2^(-1-n)*((5-sqrt(41))^n*(-7+sqrt(41)) + (5+sqrt(41))^n*(7+sqrt(41)))) / sqrt(41). - Colin Barker, Jan 20 2017
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MATHEMATICA
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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]`-$[i+1]`>4)+($[i+1]`-$[i]`>4))
(PARI) \\ Proof of generating function
TransferGf(m, u, t, v, z)=vector(m, i, u(i))*matsolve(matid(m)-z*matrix(m, m, i, j, t(i, j)), vectorv(m, i, v(i)));
RowGf(d, m, z)=1+z*TransferGf(m, i->1, (i, j)->abs(i-j)<=d, j->1, z);
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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