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A056547
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a(n) = 6*n*a(n-1) + 1 with a(0)=1.
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6
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1, 7, 85, 1531, 36745, 1102351, 39684637, 1666754755, 80004228241, 4320228325015, 259213699500901, 17108104167059467, 1231783500028281625, 96079113002205966751, 8070645492185301207085, 726358094296677108637651
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = floor(e^(1/6)*6^n*n!).
a(n) = n!*Sum_{k=0..n} 6^(n-k)/k!. E.g.f.: exp(x)/(1 - 6*x). - Philippe Deléham, Mar 14 2004
a(n) = Integral_{x = 0..inf} (6*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 6*x) satisfies the differential equation (1 - 6*x)*y' = (7 - 6*x)*y.
a(n) = (6*n + 1)*a(n-1) - 6*(n - 1)*a(n-2).
The sequence b(n) := 6^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = 6^n*n!*( 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/(6*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/6) = 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/((6*n + 1) - ... )))). Cf. A010844. (End)
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EXAMPLE
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a(2) = 6*2*a(1) + 1 = 12*7 + 1 = 85.
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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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EXTENSIONS
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STATUS
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approved
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