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A094214
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Decimal expansion of 1/phi = phi - 1.
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66
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6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, 9, 8, 0, 5, 7, 6, 2, 8, 6, 2, 1, 3, 5, 4, 4, 8, 6, 2, 2, 7, 0, 5, 2, 6, 0, 4, 6, 2, 8, 1, 8, 9, 0, 2, 4, 4, 9, 7, 0, 7, 2, 0, 7, 2, 0, 4, 1, 8, 9, 3, 9, 1, 1, 3, 7, 4, 8, 4
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OFFSET
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0,1
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COMMENTS
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Edge length of a regular decagon with unit circumradius. - Stanislav Sykora, May 07 2014
The value a+0i is the only invariant point of the complex-plane endomorphism M(z)=sqrt(2-sqrt(2+z)), and also its unique attractor, with the iterations converging exponentially from any starting complex value. Hence the infinite radical formula. - Stanislav Sykora, Apr 29 2016
With a minus sign, this constant is called beta and shares many identities with phi = A001622 (also called alpha); e.g., beta * phi = -1, Lucas numbers L(n) = A000032(n) = phi^n + beta^n. - Andrés Ventas, Apr 23 2022
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LINKS
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Eric Weisstein's World of Mathematics, Decagon.
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FORMULA
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Equals Sum_{n>=0} (15*(2*n)!-8*n!^2)/(2*n!^2*3^(2*n+2)).
Equals -1/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Equals (sqrt(5)-1)/2.
Equals 2*sin(Pi/10). (End)
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EXAMPLE
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0.6180339887498948482045868343656381177203091798057628621354486227052604628...
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MATHEMATICA
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PROG
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(PARI) default(realprecision, 20080); x=(sqrt(5)-1)/2; d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b094214.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009
(PARI)
a(n) = floor( 10^(n+1)*(quadgen(5)-1)%10);
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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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