A006277 -id:A006277 - cp's OEIS frontend

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

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

A243967 Decimal expansion of 'xi', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 16 2014

			1.3505061251311715303318376772262415972523...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, pp. 445-446.

Cf. A006277, A243968 (eta).

digits = 99; n0 = 5; dn = 5; Clear[q]; q[0] = q[1] = 1; q[n_] := q[n] = q[n - 2] (q[n - 1] + 1); xi[n_] := xi[n] = ((q[n] - 1)^(1/2*( Sqrt[5] - 1))*(q[n + 1] - 1))^((1/2*( Sqrt[5] - 1))^n/Sqrt[5]); xi[n0]; xi[n = n0 + dn]; While[RealDigits[xi[n], 10, digits + 10] != RealDigits[xi[n - 5], 10, digits + 10], Print["n = ", n];  n = n + dn]; RealDigits[xi[n], 10, digits] // First

q(n) = floor(xi^(phi^n)*eta^((1-phi)^n)) where phi is the golden ratio (1+sqrt(5))/2 and eta is A243968.

A243968 Decimal expansion of 'eta', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 16 2014

			1.42981549990099451970390644376276...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.10 Quadratic recurrence constants, pp. 445-446.

Cf. A006277, A243967 (xi).

digits = 99; n0 = 5; dn = 5; Clear[q]; q[0] = q[1] = 1; q[n_] := q[n] = q[n - 2] (q[n - 1] + 1); eta[n_] := eta[n] = ((q[n] - 1)^(-1/2 - Sqrt[5]/2)*(q[n + 1] - 1))^(-(1/((1/2*(1 - Sqrt[5]))^n*Sqrt[5]))); eta[n0]; eta[n = n0 + dn]; While[RealDigits[eta[n], 10, digits + 10] != RealDigits[eta[n - 5], 10, digits + 10], Print["n = ", n]; n = n + dn]; RealDigits[eta[n], 10, digits] // First

q(n) = floor(xi^(phi^n)*eta^((1-phi)^n)) where phi is the golden ratio (1+sqrt(5))/2.

A007704 a(n+2) = (a(n) - 1)*a(n+1) + 1.

Original entry on oeis.org

2, 3, 4, 9, 28, 225, 6076, 1361025, 8268226876, 11253255215681025, 93044467205527772332546876, 1047053135870867396062743192203958743681025

Offset: 1

N. J. A. Sloane, Jeffrey Shallit

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 = 1..18
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. A006277.

A007704 := proc(n) options remember; if n <= 2 then RETURN(n+1) else (A007704(n-2)-1)*A007704(n-1)+1; fi; end;

RecurrenceTable[{a[n] == a[n-1] (a[n-2] - 1) + 1, a[1] == 2, a[2] == 3}, a, {n, 1, 12}] (* Jean-François Alcover, Apr 05 2020 *)

a(n) = A006277(n) + 1. - R. J. Mathar, Apr 27 2007

Product_{k=1..n} a(k) = A006277(k)*A006277(k+1). - Amiram Eldar, Mar 19 2024

A141609 a(n) = (a(n-1)*a(n-2) + a(n-1)^2)/a(n-3), with a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 2, 6, 48, 1296, 290304, 1763596800, 2400297571123200, 19846204885558066176000000, 223334408639880528216369404299444224000000, 20780031060559302184531906881808103844643569442380668928000000000000

Offset: 1

Roger L. Bagula, Aug 22 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 1..18

Cf. A006277, A006720.

[n le 3 select 1 else (Self(n-1)*Self(n-2) +Self(n-1)^2)/Self(n-3): n in [1..15]]; // G. C. Greubel, Sep 21 2024

a[n_]:= a[n]= If[n<4,1,(a[n-1]*a[n-2] +a[n-1]^2)/a[n-3]]; Table[a[n], {n,15}]
RecurrenceTable[{a[1]==a[2]==a[3]==1,a[n]==(a[n-1]a[n-2]+a[n-1]^2)/a[n-3]}, a,{n,14}] (* Harvey P. Dale, Oct 01 2017 *)

def a(n): # a = A141609
    if n<3: return 1
    else: return (a(n-1)*a(n-2) +a(n-1)^2)/a(n-3)
[a(n) for n in range(1,16)] # G. C. Greubel, Sep 21 2024

a(n+1) / a(n) = A006277(n-1). - Michael Somos, Dec 29 2012

Edited by N. J. A. Sloane, Aug 24 2008

A292433 a(0) = 0, a(1) = 1; a(n) = prime(a(n-1))*a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 2, 7, 121, 79988, 81600798165, 182421074243967704954243

Offset: 0

Ilya Gutkovskiy, Dec 08 2017

			+---+-------------+--------------------+-------------------+
| n | a(n)/a(n+1) | Continued fraction |      Comment      |
+---+-------------+--------------------+-------------------+
| 1 |    1/2      | [0; 2]             |   2 = prime(a(1)) |
+---+-------------+--------------------+-------------------+
| 2 |    2/7      | [0; 3, 2]          |   3 = prime(a(2)) |
+---+-------------+--------------------+-------------------+
| 3 |    7/121    | [0; 17, 3, 2]      |  17 = prime(a(3)) |
+---+-------------+--------------------+-------------------+
| 4 |  121/79988  | [0; 661, 17, 3, 2] | 661 = prime(a(4)) |
+---+-------------+--------------------+-------------------+

Cf. A000040, A000278, A006277, A007097, A036247, A036248, A058182, A076146, A083659.

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == Prime[a[n - 1]] a[n - 1] + a[n - 2]}, a[n], {n, 7}]

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

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A243967 Decimal expansion of 'xi', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).

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A243968 Decimal expansion of 'eta', a constant related to the second order quadratic recurrence q(0)=q(1)=1, q(n)=q(n-2)*(q(n-1)+1).

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A007704 a(n+2) = (a(n) - 1)*a(n+1) + 1.

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A141609 a(n) = (a(n-1)*a(n-2) + a(n-1)^2)/a(n-3), with a(1) = a(2) = a(3) = 1.

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A292433 a(0) = 0, a(1) = 1; a(n) = prime(a(n-1))*a(n-1) + a(n-2).

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