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A190405
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Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.
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8
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6, 4, 1, 6, 3, 2, 5, 6, 0, 6, 5, 5, 1, 5, 3, 8, 6, 6, 2, 9, 3, 8, 4, 2, 7, 7, 0, 2, 2, 5, 4, 2, 9, 4, 3, 4, 2, 2, 6, 0, 6, 1, 5, 3, 7, 9, 5, 6, 7, 3, 9, 7, 4, 7, 8, 0, 4, 6, 5, 1, 6, 2, 2, 3, 8, 0, 1, 4, 4, 6, 0, 3, 7, 3, 3, 3, 5, 1, 7, 7, 5, 6, 0, 0, 3, 6, 4, 1, 7, 1, 6, 2, 3, 3, 5, 9, 1, 3, 3, 0, 8, 6, 0, 8, 9, 7, 3, 5, 3, 1, 6, 3, 4, 3, 6, 1, 9, 4, 6, 1
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OFFSET
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0,1
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COMMENTS
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This constant is not a quadratic irrational (Duverney, 1995).
The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)
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LINKS
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EXAMPLE
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0.64163256065515386629...
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MATHEMATICA
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RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Total[(1/2)^Accumulate[Range[50]]], 10, 120][[1]] (* Harvey P. Dale, Oct 18 2013 *)
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PROG
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(Sage)
def A190405(b): # Generate the constant with b bits of precision
return N(sum([(1/2)^(j*(j+1)/2) for j in range(1, b)]), b)
(PARI) th2(x)=2*x^.25 + 2*suminf(n=1, x^(n+1/2)^2)
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CROSSREFS
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A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T = A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T = A000217 (triangular numbers).
A190406: Sum_{k>=1} (1/2)^S(k-1), where S = A001844 (centered square numbers).
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KEYWORD
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AUTHOR
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STATUS
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approved
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