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A307469
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a(n) = 2*a(n-1) + 6*a(n-2) for n >= 2, a(0) = 1, a(1) = 5.
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0
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1, 5, 16, 62, 220, 812, 2944, 10760, 39184, 142928, 520960, 1899488, 6924736, 25246400, 92041216, 335560832, 1223368960, 4460102912, 16260419584, 59281456640, 216125430784, 787939601408, 2872631787520, 10472901183488, 38181593092096, 139200593285120
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of words of length n over alphabet {1,2,3,4,5} such that no odd letter is followed by an odd letter.
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LINKS
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FORMULA
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a(n) = (-(2/7)*sqrt(7)+1/2)*(1-sqrt(7))^n+((2/7)*sqrt(7)+1/2)*(1+sqrt(7))^n.
G.f.: (1+3*x)/(1-2*x-6*x^2).
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EXAMPLE
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For n=2 the a(2)=16 solutions are: 12, 14, 21, 22, 23, 24, 25, 32, 34, 41, 42, 43, 44, 45, 52, 54.
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MAPLE
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aseq:=proc(n) option remember;
if n<0 then return "seq not defined for negative indices";
elif n=0 then return 1;
elif n=1 then return 5;
else 2*aseq(n-1)+6*aseq(n-2);
end if;
end proc:
seq(aseq(n), n=0..26);
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MATHEMATICA
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a[0] = 1; a[1] = 5;
a[n_] := a[n] = 2*a[n - 1] + 6*a[n - 2];
Table[a[n], {n, 0, 26}]
LinearRecurrence[{2, 6}, {1, 5}, 30] (* Harvey P. Dale, Feb 20 2023 *)
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CROSSREFS
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The same over alphabet {1,2,3} gives A001045(n+2).
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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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