

A093540


Decimal expansion of Sum_{n >= 1} 1/L(n), where L(n) is the nth Lucas number.


14



1, 9, 6, 2, 8, 5, 8, 1, 7, 3, 2, 0, 9, 6, 4, 5, 7, 8, 2, 8, 6, 8, 7, 9, 5, 1, 2, 8, 6, 7, 5, 1, 8, 3, 5, 2, 6, 6, 4, 9, 5, 9, 3, 0, 1, 7, 1, 6, 2, 2, 1, 9, 4, 2, 1, 1, 3, 0, 7, 1, 5, 2, 4, 0, 4, 1, 7, 0, 6, 1, 6, 0, 7, 5, 4, 6, 4, 6, 0, 3, 7, 7, 9, 7, 9, 0, 4, 1, 8, 9, 9, 0, 8, 4, 0, 3, 4, 6, 9, 6, 2, 2
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OFFSET

1,2


COMMENTS

AndréJeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)).  Amiram Eldar, Oct 30 2020


LINKS

Paul S. Bruckman, Problem B603, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 280; Lucas Analogue, Solution to Problem B603 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), p. 282.


FORMULA

Equals Sum_{k>=0} 1/(phi^(2*k+1)  (1)^k), where phi is the golden ratio (A001622).
Equals 7/3  10 * Sum_{k>=1} 1/(L(2*k1)*L(2*k+1)*L(2*k+2)) (Bruckman, 1987).  Amiram Eldar, Jan 27 2022


EXAMPLE

1.96285817320964578286879512867518352664959301716221...


MATHEMATICA

RealDigits[Sum[1/LucasL[n], {n, 2000}], 10, 120][[1]] (* Harvey P. Dale, Jan 15 2012 *)


PROG



CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



