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Decimal expansion of Sum_{k>=0} 1/A007018(k).
1

%I #7 Mar 19 2024 10:29:01

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

%T 7,9,3,8,0,8,5,7,7,2,4,2,5,7,9,5,2,4,9,4,4,9,6,0,4,6,6,0,5,4,0,0,0,0,

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

%N Decimal expansion of Sum_{k>=0} 1/A007018(k).

%C The corresponding alternating sum, Sum_{k>=0} (-1)^k/A007018(k), equals Cahen's constant (A118227).

%C Duverney et al. (2018) proved that this constant is transcendental.

%C Called the "Kellogg-Curtiss constant" by Sondow (2021), after the American mathematicians Oliver Dimon Kellogg (1878-1932) and David Raymond Curtiss (1878-1953).

%C The Engel expansion of this constant is 1 followed by the Sylvester sequence (A000058, see the Formula section).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7, p. 436.

%H Brenda S. Baker and Edward G. Coffman, Jr., <a href="https://doi.org/10.1137/0602019">A tight asymptotic bound for next-fit-decreasing bin-packing</a>, SIAM Journal on Algebraic Discrete Methods, Vol. 2, No. 2 (1981), pp. 147-152.

%H Daniel Duverney, Takeshi Kurosawa, and Iekata Shiokawa, <a href="https://doi.org/10.2140/moscow.2019.8.57">Transcendence of numbers related with Cahen's constant</a>, Moscow Journal of Combinatorics and Number Theory, Vol. 8, No. 1 (2018), pp. 57-69; <a href="https://web.archive.org/web/20190226045451id_/http://pdfs.semanticscholar.org/7949/9fc36df73bb2d9d87fb9c08d08bf4770cb7c.pdf">alternative link</a>.

%H Daniel Duverney, Takeshi Kurosawa, and Iekata Shiokawa. <a href="https://doi.org/10.21099/tkbjm/20204402235">Irrationality exponents of certain fast converging series of rational numbers</a>, Tsukuba Journal of Mathematics, Vol. 44, No. 2 (2020), pp. 235-250; <a href="https://danielduverney.fr/documents/theorie-des-nombres/Tsukuba.pdf">alternative link</a>.

%H Chan C. Lee and Der-Tsai Lee, <a href="https://doi.org/10.1145/3828.3833">A simple on-line bin-packing algorithm</a>, Journal of the ACM (JACM), Vol. 32, No. 3 (1985), pp. 562-572. See p. 566.

%H Iekata Shiokawa, <a href="https://cir.nii.ac.jp/crid/1050568617162367872">Irrationality exponents of certain alternating serries</a>, Analytic Number Theory and Related Topics, Vol. 2162 (2020), pp. 210-215.

%H Jonathan Sondow, <a href="https://doi.org/10.1007/978-3-030-84304-5_22">Irrationality and Transcendence of Alternating Series via Continued Fractions</a>, in: A. Bostan and K. Raschel (eds.), Transcendence in Algebra, Combinatorics, Geometry and Number Theory, TRANS 2019. Springer Proceedings in Mathematics & Statistics, Vol. 373, Springer, Cham, 2021; <a href="https://arxiv.org/abs/2009.14644">arXiv preprint</a>, arXiv:2009.14644 [math.NT], 2020.

%H Andrew Twigg and Eduardo C. Xavier, <a href="https://doi.org/10.1016/j.tcs.2015.06.036">Locality-preserving allocations problems and coloured bin packing</a>, Theoretical Computer Science, Vol. 596 (2015), pp. 12-22.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bin_packing_problem">Bin packing problem</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Harmonic_bin_packing">Harmonic bin packing</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Next-fit-decreasing_bin_packing">Next-fit-decreasing bin packing</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals 1 + Sum_{k>=1} 1/(Product_{i=0..k-1} A000058(i)).

%e 1.69103020675725397443566284314574179380857724257952...

%t s[0] = 2; s[n_] := s[n] = s[n - 1]^2 - s[n - 1] + 1; kmax = 1; FixedPoint[RealDigits[Sum[1/(s[k] - 1), {k, 0, kmax += 10}], 10, 120][[1]] &, kmax] (* after _Jean-François Alcover_ at A118227 *)

%o (PARI) c = 1; 1 + suminf(k = 1, c += c^2; 1/c) \\ after _Charles R Greathouse IV_ at A118227

%Y Cf. A000058, A007018, A118227, A242724.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Mar 19 2024