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A143820
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Decimal expansion of the constant 1/1! + 1/4! + 1/7! + ...
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11
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1, 0, 4, 1, 8, 6, 5, 3, 5, 5, 0, 9, 8, 9, 0, 9, 8, 4, 6, 3, 0, 1, 3, 3, 6, 6, 1, 5, 0, 2, 1, 5, 2, 7, 3, 8, 7, 6, 9, 7, 0, 8, 3, 5, 7, 1, 7, 2, 4, 1, 6, 3, 4, 5, 9, 5, 4, 5, 7, 3, 9, 2, 5, 5, 4, 2, 3, 5, 5, 1, 7, 4, 1, 1, 6, 1, 0, 7, 4, 0, 2, 9, 5, 9, 2, 8, 6, 2, 6, 7, 3, 9, 3, 0, 1, 0, 0, 6, 5, 5, 2
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OFFSET
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1,3
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COMMENTS
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Define a sequence R(n) of real numbers by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(2) - R(1); the decimal expansions of R(0) = 1 + 1/3! + 1/6! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1).
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
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| linear combination of
R(n) | R(0) R(1) R(2) - R(1)
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R(3) | 1 1 3
R(4) | 6 2 7
R(5) | 25 11 16
R(6) | 91 66 46
R(7) | 322 352 203
R(8) | 1232 1730 1178
R(9) | 5672 8233 7242
R(10) | 32202 39987 43786
...
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LINKS
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FORMULA
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Equals (exp(1) + w^2*exp(w) + w*exp(w^2))/3, where w = exp(2*Pi*i/3).
Continued fraction: 1 + 1/(24 - 24/(211 - 210/(721 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n - 1)*(3*n)*(3*n + 1) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024
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EXAMPLE
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1.041865355098909...
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MAPLE
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Digits:=101: evalf(sum(1/(3*n+1)!, n=0..infinity)); # Michal Paulovic, Aug 20 2023
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MATHEMATICA
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RealDigits[ N[ (-Cos[Sqrt[3]/2] + E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E]), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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