A242724 -id:A242724 - cp's OEIS frontend

A118227 Decimal expansion of Cahen's constant.

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

Offset: 0

Eric W. Weisstein, Apr 16 2006

Cahen proved that his constant is irrational. Davison and Shallit proved that it is transcendental and computed its simple continued fraction expansion A006280. - Jonathan Sondow, Aug 17 2014

Named after the French mathematician Eugène Cahen (1865 - 1941). - Amiram Eldar, Oct 29 2020

			0.6434105462883380261...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.7, p. 436.

Eugène Cahen, Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues, Nouvelles Annales de Mathématiques, Vol. 10 (1891), pp. 508-514.
J. L. Davison and Jeffrey Shallit, Continued Fractions for Some Alternating Series, Monatsh. Math., Vol. 111, No. 2 (1991), pp. 119-126, alternative link.
Eric Weisstein's World of Mathematics, Cahen's Constant.
Wikipedia, Cahen's constant.
Index entries for transcendental numbers.

Cf. A000058, A006279, A006280, A006281, A123180, A129871, A242724.

a[0] = 2; a[n_] := a[n] = a[n-1]^2 - a[n-1]+1; kmax = 1; FixedPoint[ RealDigits[ Sum[(-1)^k/(a[k]-1), {k, 0, kmax += 10}], 10, 105][[1]]&, kmax] (* Jean-François Alcover, Jul 28 2011, updated Jun 19 2014 *)
Most@First@RealDigits@N[x=1; 1+Sum[x=x(1+x); (-1)^k/x, {k, 1, 9}], 106] (* Oliver Seipel, Aug 25 2024, after Charles R Greathouse IV *)
Most@First@RealDigits@N[x=1; 1/2+Sum[x=x(1+x)(1+x+x^2); 1/(x+1), {k, 1, 4}], 106] (* Oliver Seipel, Aug 25 2024 *)

C=1;1+suminf(k=1,C+=C^2; (-1)^k/C) \\ Charles R Greathouse IV, Jul 14 2020

Equals Sum_{k >= 0} (-1)^k/(A000058(k)-1).
Equals Sum_{n>=0} 1/A000058(2*n) = 1 - Sum_{n>=0} 1/A000058(2*n+1). - Amiram Eldar, Oct 29 2020
Equals 1 + (1/2) * Sum_{n>=0} (-1)^(n+1)/A129871(n). - Bernard Schott, Apr 06 2021

A006277 a(n) = (a(n-1) + 1)*a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 8, 27, 224, 6075, 1361024, 8268226875, 11253255215681024, 93044467205527772332546875, 1047053135870867396062743192203958743681024, 97422501162981936223682742789520433197690551802305989766350860546875

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.7.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7 Cahen's Constant p. 435 and Section 6.10 Quadratic recurrence constants pp. 445-446.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harvey P. Dale, Table of n, a(n) for n = 0..18
J. L. Davison and Jeffrey Shallit, Continued Fractions for Some Alternating Series, Monatsh. Math., Vol. 111 (1991), pp. 119-126.

Cf. A001622, A242724, A243967, A243968.

a006277_list = 1 : scanl ((*) . (+ 1)) 2 a006277_list -- Jack Willis, Dec 22 2013

[n le 2 select 1 else (Self(n-1) + 1)*Self(n-2): n in [1..15]]; // Vincenzo Librandi, May 23 2019

A006277 := proc(n) options remember; if n <= 1 then RETURN(1) else A006277(n-2)*(A006277(n-1)+1); fi; end;

a=b=1;lst={a,b};Do[AppendTo[lst,c=a*b+a];a=b;b=c,{n,0,12}];lst (* Vladimir Joseph Stephan Orlovsky, May 06 2010 *)
RecurrenceTable[{a[n]==a[n-2]*(1+a[n-1]),a[0]==1,a[1]==1},a,{n,0,15}] (* Vaclav Kotesovec, Jan 19 2015 *)
nxt[{a_,b_}]:={b,a(b+1)}; NestList[nxt,{1,1},15][[All,1]] (* Harvey P. Dale, Jun 20 2021 *)

a(n) := if (n = 0 or n = 1) then 1 else a(n-2)*(a(n-1)+1) $
makelist(a(n),n,0,12); /* Emanuele Munarini, Mar 23 2017 */

Sum_{n>=0} 1/a(n) = 3. - Gerald McGarvey, Jul 20 2004
a(n) = floor(A243967^(phi^n) * A243968^((1-phi)^n)), where phi is the golden ratio (1+sqrt(5))/2. - Vaclav Kotesovec, Jan 19 2015
Sum_{k>=0} (-1)^k/(a(k)*a(k+1)) = A242724. - Amiram Eldar, May 15 2021

More terms from Vladimir Joseph Stephan Orlovsky, May 06 2010

A371321 Decimal expansion of Sum_{k>=0} 1/A007018(k).

Original entry on oeis.org

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, 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, 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

Offset: 1

Amiram Eldar, Mar 19 2024

The corresponding alternating sum, Sum_{k>=0} (-1)^k/A007018(k), equals Cahen's constant (A118227).
Duverney et al. (2018) proved that this constant is transcendental.
Called the "Kellogg-Curtiss constant" by Sondow (2021), after the American mathematicians Oliver Dimon Kellogg (1878-1932) and David Raymond Curtiss (1878-1953).
The Engel expansion of this constant is 1 followed by the Sylvester sequence (A000058, see the Formula section).

			1.69103020675725397443566284314574179380857724257952...

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

Brenda S. Baker and Edward G. Coffman, Jr., A tight asymptotic bound for next-fit-decreasing bin-packing, SIAM Journal on Algebraic Discrete Methods, Vol. 2, No. 2 (1981), pp. 147-152.
Daniel Duverney, Takeshi Kurosawa, and Iekata Shiokawa, Transcendence of numbers related with Cahen's constant, Moscow Journal of Combinatorics and Number Theory, Vol. 8, No. 1 (2018), pp. 57-69; alternative link.
Daniel Duverney, Takeshi Kurosawa, and Iekata Shiokawa. Irrationality exponents of certain fast converging series of rational numbers, Tsukuba Journal of Mathematics, Vol. 44, No. 2 (2020), pp. 235-250; alternative link.
Chan C. Lee and Der-Tsai Lee, A simple on-line bin-packing algorithm, Journal of the ACM (JACM), Vol. 32, No. 3 (1985), pp. 562-572. See p. 566.
Iekata Shiokawa, Irrationality exponents of certain alternating serries, Analytic Number Theory and Related Topics, Vol. 2162 (2020), pp. 210-215.
Jonathan Sondow, Irrationality and Transcendence of Alternating Series via Continued Fractions, 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; arXiv preprint, arXiv:2009.14644 [math.NT], 2020.
Andrew Twigg and Eduardo C. Xavier, Locality-preserving allocations problems and coloured bin packing, Theoretical Computer Science, Vol. 596 (2015), pp. 12-22.
Wikipedia, Bin packing problem.
Wikipedia, Harmonic bin packing.
Wikipedia, Next-fit-decreasing bin packing.
Index entries for transcendental numbers.

Cf. A000058, A007018, A118227, A242724.

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 *)

c = 1; 1 + suminf(k = 1, c += c^2; 1/c) \\ after Charles R Greathouse IV at A118227

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

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A118227 Decimal expansion of Cahen's constant.

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A006277 a(n) = (a(n-1) + 1)*a(n-2).

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A371321 Decimal expansion of Sum_{k>=0} 1/A007018(k).

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