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A274520
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a(n) = ((1 + sqrt(7))^n - (1 - sqrt(7))^n)/sqrt(7).
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3
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0, 2, 4, 20, 64, 248, 880, 3248, 11776, 43040, 156736, 571712, 2083840, 7597952, 27698944, 100985600, 368164864, 1342243328, 4893475840, 17840411648, 65041678336, 237125826560, 864501723136, 3151758405632, 11490527150080, 41891604733952, 152726372368384
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OFFSET
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0,2
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COMMENTS
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Number of zeros in substitution system {0 -> 111, 1 -> 1001} at step n from initial string "1" (see example).
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LINKS
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FORMULA
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O.g.f.: 2*x/(1 - 2*x - 6*x^2).
E.g.f.: 2*exp(x)*sinh(sqrt(7)*x)/sqrt(7).
Dirichlet g.f.: (PolyLog(s,1+sqrt(7)) - PolyLog(s,1-sqrt(7)))/sqrt(7), where PolyLog(s,x) is the polylogarithm function.
a(n) = 2*a(n-1) + 6*a(n-2).
Lim_{n->infinity} a(n+1)/a(n) = 1 + sqrt(7) = 1 + A010465.
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EXAMPLE
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Evolution from initial string "1": 1 -> 1001 -> 10011111111001 -> 1001111111100110011001100110011001100110011111111001 -> ...
Therefore, number of zeros at step n:
a(0) = 0;
a(1) = 2;
a(2) = 4;
a(3) = 20, etc.
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MATHEMATICA
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LinearRecurrence[{2, 6}, {0, 2}, 27]
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PROG
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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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