A006280 -id:A006280 - 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

A006281 Partial quotients in continued fraction expansion of 2C-1, where C is Cahen's constant.

Original entry on oeis.org

0, 3, 2, 18, 98, 33282, 319994402, 354455304050635218, 36294953231792713902640647988908098

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..12
J. L. Davison and Jeffrey O. Shallit, Continued Fractions for Some Alternating Series, Monatshefte für Mathematik, Vol. 111 (1991), pp. 119-126; alternative link.

Cf. A006279, A006280, A118227.

s[0] = 2; s[n_] := s[n] = s[n - 1]^2 - s[n-1] + 1; h[0] = 1; h[n_] := h[n] = (s[n] - 1)/(2*h[n-1]); a[0] = 0; a[1] = 3; a[n_] := 2*h[n-1]^2; Array[a, 9, 0] (* Amiram Eldar, Mar 19 2024 *)

A006279 Denominators of convergents to Cahen's constant: a(n+2) = a(n)^2*a(n+1) + a(n).

Original entry on oeis.org

1, 1, 2, 3, 14, 129, 25298, 420984147, 269425140741515486, 47749585090209528873482531562977121, 3466137915373323052799848584927709551269254572949111609037058632767202

Offset: 0

N. J. A. Sloane

Shifted square roots of partial quotients in continued fraction expansion of Cahen's constant: a(n) = sqrt(A006280(n+2)). - Jonathan Sondow, Aug 20 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..13
J. L. Davison and Jeffrey O. Shallit, Continued Fractions for Some Alternating Series, Monatshefte für Mathematik, Vol. 111 (1991), pp. 119-126; alternative link.

Cf. A118227, A006280, A006281.

A006279 := proc(n) option remember; if n <= 1 then 1 else A006279(n-2)^2*A006279(n-1)+A006279(n-2) fi end:
seq(A006279(n), n=0..10);

a[n_] := a[n] = If[n < 2, 1, a[n-2]^2*a[n-1] + a[n-2]];
Table[a[n], {n, 0, 9}] (* Jean-François Alcover, Sep 23 2022 *)

from itertools import islice
def A006279_gen(): # generator of terms
    a, b = 1, 1
    yield a
    while True:
        yield b
        a, b = b, a*(a*b+1)
A006279_list = list(islice(A006279_gen(),10)) # Chai Wah Wu, Mar 19 2024

Definition clarified by Jonathan Sondow, Aug 20 2014

A123180 Even positions of Sylvester's sequence A000058; the denominators of the (greedy) Egyptian fraction expansion of Cahen's constant.

Original entry on oeis.org

2, 7, 1807, 10650056950807, 12864938683278671740537145998360961546653259485195807

Offset: 0

David Eppstein, Oct 03 2006

Amiram Eldar, Table of n, a(n) for n = 0..6
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.
Eric Weisstein's World of Mathematics, Cahen's Constant.
Wikipedia, Cahen's constant.

Cf. A000058, A006279, A006280, A006281, A077125, A118227.

f[n_] := n*(n-1)*(n*(n-1)+1)+1; a[0] = 2; a[n_] := a[n] = f[a[n-1]]; Array[a, 5, 0] (* Amiram Eldar, Mar 19 2024 *)
1+NestList[#(#+1)(#^2+#+1) &, 1, 4] (* Oliver Seipel, Aug 25 2024 *)

a(n)=if(n, my(k=a(n-1));k*=k-1; k*(k+1)+1, 2) \\ Charles R Greathouse IV, Dec 12 2013

a(n) = a(n-1)*(a(n-1)-1)*(a(n-1)*(a(n-1)-1)+1)+1.
a(n) is approximately k^4^n with k = 1.5979102180318731783... (A077125). - Charles R Greathouse IV, Dec 12 2013
Sum_{n>=0} 1/a(n) = A118227. - Amiram Eldar, Mar 19 2024

a(4) from Charles R Greathouse IV, Dec 12 2013

A242724 Decimal expansion of a constant associated with self-generating continued fractions and Cahen's constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 21 2014

This constant is known to be transcendental.

Called the "Davison-Shallit constant" by Finch (2003) and Sondow (2021). - Amiram Eldar, Mar 19 2024

			0.62946502045518677183129422910723212269353...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.7, p. 435.

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. In French.
J. L. Davison and Jeffrey O. Shallit, Continued Fractions for Some Alternating Series, Monatshefte für Mathematik, Vol. 111 (1991), pp. 119-126; alternative link.
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.
Eric Weisstein's MathWorld, Cahen's constant.
Wikipedia, Cahen's constant.
Index entries for transcendental numbers.

Cf. A006277, A006279, A006280, A006281, A118227, A123180.

digits = 100; Clear[q, s]; q[n_] := q[n] = q[n - 2]*(q[n-1] + 1); q[0] = q[1] = 1; s[k_] := s[k] = Sum[(-1)^j/(q[j]*q[j+1]), {j, 0, k}] // N[#, digits+5]&; s[dk = 5]; s[k = 2*dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits], Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First

Equals Sum_{k>=0} (-1)^k/(A006277(k)*A006277(k+1)). - Amiram Eldar, Mar 19 2024

A380013 Continued fraction expansion of Sum_{i>=0} (-1)^i/(q(i)q(i+1)) where q(0)=q(1)=1, q(2n+2)=q(2n+1)+q(2n), and q(2n+3)=q(2n+1)(q(2n+2)+1).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 18, 1, 432, 1, 196992, 1, 38895676416, 1, 1512881323731695591424, 1, 2288809899755012359448064967916189926490112, 1

Offset: 0

Khalil Ayadi, Jan 09 2025

a(19) has 85 decimal digits and a(21) has 170 decimal digits.
This number is transcendental.
q(n) is the denominator of the convergent resulting from terms a(0..n).
The continued fraction is constructed by successively appending a pair of terms 1 and its own q(n) so far, so a(2*n) = 1 and a(2*n+1) = q(2*n-1) for n>=1
The series and the recurrence for q follows from that construction.
The series can also be written Sum_{i>=0} (-1)^i/x(i) where x(i) = q(i)*q(i+1) and in that case x(0)=1, x(2n+1) divides x(2n+2), and x(2n+3) = ((x(2n+2)/x(2n+1))*(x(2n)/x(2n-1))*...*(x(2)/x(1)))^2 + x(2n+2).

			0 + 1/(1 + 1/(1 + 1/(1 + ... ))) = 0.6087912199223083952132365...

Andrew Howroyd, Table of n, a(n) for n = 0..25
Khalil Ayadi, Chiheb Ben Bechir, and Maher Saadaoui, Continued Fractions with Predictable Patterns and Transcendental Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.4.

Cf. A006280, A019426.

Q(n) = {my(v=vector(n+1)); v[1]=v[2]=1; for(i=2, n, v[i+1] = if(i%2==0, v[i]+v[i-1], v[i-1]*(v[i]+1))); v}
seq(n)=my(q=Q(max(2,n-2))); vector(n+1, n, if(n%2 || n<4, n>1, q[n-2])) \\ Andrew Howroyd, Jan 13 2025

A380194 Continued fraction expansion of Sum_{i>=0} (-1)^i/(q(i)q(i+1)) where q(0)=q(1)=1, q(3n+2)=q(3n+1)+q(3n), q(3n+3)=q(3n+2)+q(3n+1), and q(3n+4)=q(3n+2)(q(3n+2)*q(3n+3)+1).

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 1, 289, 1, 1, 81126049, 1, 1, 2128359349797626142548649, 1, 1, 38565134716822109850786884343127955049217538196275147632486387905655060249, 1, 1

Offset: 0

Khalil Ayadi, Jan 15 2025

This is a transcendental number.
The n-th convergent of a(0..n) has q(n) as denominator.
Thus a(3*n+2) = a(3*n+3)=1 and a(3*n+4) = q(3*n+2)^2 for n>=1 are the results of repeatedly appending a triple of terms 1,1,Q^2 where Q is the convergent denominator after the first new 1.
The recurrence for q follows from this construction, and the alternating series is the continued fraction value for any sequence of convergent denominators.
This structure leads to the series and the recurrence for q.
Sum_{i>=0} (-1)^i/x(i) is another way to write the series, where x(i) = q(i)*q(i+1). When x(0)=1 , x(3n+2) divides x(3n+3), x(3n+2)-x(3n+1)=((x(3n+1))/x(3n))*(x(3n-1)/x(3n-2))*(x(3n-3)/x(3n-4))...(x(2)/x(1)))^2,x(3n+4)-x(3n+3)=(x(3n+3)/x(3n+2))^2*(x(3n+2)-x(3n+1)).

			0 + 1/(1 + 1/(1 + 1/(1 + ... ))) = 0.645164877940276...

Khalil Ayadi, Chiheb Ben Bechir, and Maher Saadaoui, Continued Fractions with Predictable Patterns and Transcendental Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.4.

Cf. A006280, A019426, A003417.

q(n) = if (n<=1, 1, if (n%3==1, q(n-2)*(q(n-2)*q(n-1)+1), q(n-1)+q(n-2)));
a(n) = if (n==0, 0, if ((n%3)==1, q(n-2)^2, 1)); \\ Michel Marcus, Jan 17 2025

cp's OEIS Frontend

A118227 Decimal expansion of Cahen's constant.

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A006281 Partial quotients in continued fraction expansion of 2C-1, where C is Cahen's constant.

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A006279 Denominators of convergents to Cahen's constant: a(n+2) = a(n)^2*a(n+1) + a(n).

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A123180 Even positions of Sylvester's sequence A000058; the denominators of the (greedy) Egyptian fraction expansion of Cahen's constant.

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A242724 Decimal expansion of a constant associated with self-generating continued fractions and Cahen's constant.

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A380013 Continued fraction expansion of Sum_{i>=0} (-1)^i/(q(i)*q(i+1)) where q(0)=q(1)=1, q(2n+2)=q(2n+1)+q(2n), and q(2n+3)=q(2n+1)*(q(2n+2)+1).

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A380194 Continued fraction expansion of Sum_{i>=0} (-1)^i/(q(i)*q(i+1)) where q(0)=q(1)=1, q(3n+2)=q(3n+1)+q(3n), q(3n+3)=q(3n+2)+q(3n+1), and q(3n+4)=q(3n+2)*(q(3n+2)*q(3n+3)+1).

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A380013 Continued fraction expansion of Sum_{i>=0} (-1)^i/(q(i)q(i+1)) where q(0)=q(1)=1, q(2n+2)=q(2n+1)+q(2n), and q(2n+3)=q(2n+1)(q(2n+2)+1).

A380194 Continued fraction expansion of Sum_{i>=0} (-1)^i/(q(i)q(i+1)) where q(0)=q(1)=1, q(3n+2)=q(3n+1)+q(3n), q(3n+3)=q(3n+2)+q(3n+1), and q(3n+4)=q(3n+2)(q(3n+2)*q(3n+3)+1).