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A094885
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Decimal expansion of phi*e, where phi = (1 + sqrt(5))/2.
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9
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4, 3, 9, 8, 2, 7, 2, 3, 8, 9, 4, 4, 7, 9, 4, 6, 3, 9, 5, 9, 7, 1, 9, 8, 7, 0, 2, 9, 2, 9, 2, 8, 8, 5, 8, 6, 8, 7, 8, 6, 7, 4, 0, 4, 9, 7, 9, 7, 8, 8, 3, 4, 9, 1, 7, 0, 3, 8, 0, 9, 8, 0, 9, 0, 2, 1, 6, 4, 4, 4, 4, 3, 2, 1, 1, 6, 2, 0, 4, 4, 3, 0, 0, 3, 8, 5, 4, 6, 4, 3, 5, 2, 9, 2, 9, 4, 7, 2, 6
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OFFSET
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1,1
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COMMENTS
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Matches the value of the infinite nested radical corresponding to the sequence {e^(2^n), n=1,2,3,...}, i.e., a = sqrt(e^2+sqrt(e^4+...)), which converges by Vijayaraghavan's theorem. Proof: write the golden ratio as phi = sqrt(1+ sqrt(1+ sqrt(1+...))). Then e*phi = e*sqrt(1+ sqrt(1+ sqrt(1+ ...))) = sqrt(e^2+ e^2*sqrt(1+ sqrt(1+ ...))) = sqrt(e^2+ sqrt(e^4+ e^4*sqrt(1+ ...))) = ... = a. Evidently, the 'e' could stand for any constant, not just e; for example phi itself as in A104457, or Pi as in A094886. - Stanislav Sykora, May 24 2016
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LINKS
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EXAMPLE
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4.398272389447946...
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MATHEMATICA
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PROG
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(PARI) default(realprecision, 20080); phi=(1+sqrt(5))/2; x=phi*exp(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b094885.txt", n, " ", d)); \\ Harry J. Smith, Apr 27 2009
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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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