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Search: id:A105393
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| A105393 |
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Decimal expansion of sum of reciprocals of squares of Fibonacci numbers. |
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+0
2
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| 2, 4, 2, 6, 3, 2, 0, 7, 5, 1, 1, 6, 7, 2, 4, 1, 1, 8, 7, 7, 4, 1, 5, 6, 9, 4, 1, 2, 9, 2, 6, 6, 2, 0, 3, 7, 4, 3, 2, 0, 2, 5, 9, 7, 7, 4, 5, 1, 3, 8, 3, 0, 9, 0, 5, 1, 1, 0, 1, 0, 2, 8, 3, 4, 5, 4, 6, 6, 1, 1, 9, 3, 7, 5, 1, 1, 1, 9, 7, 8, 6, 3, 6, 8, 7, 7, 5, 3, 8, 9, 8, 1, 5, 2, 1, 5, 3, 6, 3, 6, 3, 7, 9, 2, 1
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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This is one of a set of related number theory constants. It is conjectured to be irrational. T. D. Noe (noe(AT)sspectra.com) extended the digits of this constant to 100 digits accuracy, when J. V. Post had discovered it and found it to 8 digits. The reciprocal Fibonacci constant sum 1/F(n) ~ 3.35988566 is given in A079586, queried as to irrationality by Erdos and proved irrational by Andre-Jeannin (1989). The reciprocal Lucas constant sum 1/L(n) ~ 1.96285817 is given in A093540. Sum 1/F(n)^2 converges because each term is equal or less than the corresponding term in the converging reciprocal Fibonacci constant. The sum 1/L(n)^2 is given in A105394.
Known to be transcendental. - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 07 2006
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REFERENCES
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R. Andre-Jeannin, "Irrationalite de la somme des inverses de certaines suites recurrentes." C. R. Acad. Sci. Paris Ser. I Math. 308, 539-541, 1989.
Michel Waldschmidt, Elliptic functions and transcendance, Dec. 2005, to appear
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LINKS
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Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
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FORMULA
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Decimal expansion of Sum 1/F(n)^2.
Sum(k>=1, 1/F(k)^2) = 2.4263207511672411877... - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 07 2006
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EXAMPLE
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2.426320751167241187741569...
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PROGRAM
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(PARI) sum(k=1, 500, 1./fibonacci(k)^2) (Cloitre)
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CROSSREFS
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Cf. A079586, A093540, A105394.
Sequence in context: A010241 A162630 A102128 this_sequence A049200 A164701 A083791
Adjacent sequences: A105390 A105391 A105392 this_sequence A105394 A105395 A105396
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KEYWORD
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cons,easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 04 2005
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EXTENSIONS
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More terms from Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 07 2006
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