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A339237
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Decimal expansion of K = Sum_{m>=0} 1/(1 + 2*m + 4*m^2).
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2
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1, 2, 7, 9, 7, 2, 8, 7, 4, 2, 2, 8, 1, 8, 9, 6, 8, 3, 3, 6, 4, 7, 2, 7, 5, 7, 0, 1, 5, 0, 7, 6, 3, 0, 6, 7, 2, 2, 6, 2, 6, 0, 3, 6, 7, 5, 0, 7, 5, 7, 8, 2, 6, 1, 9, 3, 0, 6, 8, 3, 0, 5, 8, 8, 1, 6, 9, 3, 0, 6, 6, 0, 7, 2, 2, 1, 3, 6, 4, 9, 0, 6, 6, 2, 1, 1, 5, 3, 2, 9, 9, 0, 5, 3, 5, 3, 2, 2, 7, 3, 7, 1, 9, 7, 1, 3, 2, 9, 2, 3
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OFFSET
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1,2
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COMMENTS
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This constant K and the constant J = A339135 allow the expression of the real and imaginary parts of:
Psi(1/4 + i*sqrt(3)/4) = - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*sqrt(3)*K;
Psi(-1/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3) - sqrt(3)*K + Pi*tanh(Pi*sqrt(3)/2));
Psi(3/4 + i*sqrt(3)/4)= - J - i*sqrt(3)*k - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/4) + i*Pi*tanh(Pi*sqrt(3)/2).
Psi(-3/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3)/3 + sqrt(3)*K).
where Psi is the digamma function and i=sqrt-1).
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LINKS
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FORMULA
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Equals -i/(2*sqrt(3)) * (Psi(1/4 + i*sqrt(3)/4) - Psi(1/4 - i*sqrt(3)/4)).
Equals Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/3 - Sum_{m>=0} 1/(3 + 6*m + 4*m^2).
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EXAMPLE
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1.27972874228189683364727570150763067226260...
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MAPLE
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K:= Re(sum(1/(1+2*n+4*n^2), n=0..infinity)):
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MATHEMATICA
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RealDigits[N[Re[Sum[1/(1 + 2*n + 4*n^2), {n, 0, Infinity}]], 110]][[1]]
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PROG
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(PARI) sumpos(n=0, 1/(1+2*n+4*n^2)) \\ Michel Marcus, Nov 28 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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