A154310 - OEIS

A154310

Decimal expansion of convergent sum of weighted self-defining reciprocals.

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

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OFFSET

1,2

COMMENTS

Let x(1)=1, let x(2) be x>0 satisfying 1/x(1) + 1/x = x,..., for n>=1, let x(n+1) be x>0 satisfying SUM(1/x(k): k=1,2,...,n) + 1/x = x.

Then SUM(1/x(k): k=1,2,...) diverges, but if the k-th term is replaced by w(k)/x(k) where w(k)=2^(1-k), the resulting sum converges to S=1.518737... .

x(1)=1, x(2)=1.618... (the golden ratio), x(3)=2.095293... .

LINKS

Table of n, a(n) for n=1..84.

Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.

EXAMPLE

1.51873724747790391474429875017680513439652335339033...

CROSSREFS

Sequence in context: A201525 A269229 A193089 * A193856 A255294 A115521

Adjacent sequences: A154307 A154308 A154309 * A154311 A154312 A154313

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling and Peter J. C. Moses, Jan 06 2009

STATUS

approved