A028257 -id:A028257 - cp's OEIS frontend

A220335 A modified Engel expansion for sqrt(3) - 1.

2, 3, 4, 2, 8, 14, 2, 98, 194, 2, 18818, 37634, 2, 708158978, 1416317954, 2, 1002978273411373058, 2005956546822746114, 2, 2011930833870518011412817828051050498, 4023861667741036022825635656102100994

Offset: 1

Peter Bala, Dec 12 2012

This is the case p = 2 of a family of quadratic irrationals of the form (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1) whose modified Engel expansion, as defined below, has a predictable form. For other cases see A220336 (p = 3), A220337 (p = 4) and A220338 (p = 5).
The Engel expansion of a positive real number x in the half-open interval (0,1] is the unique nondecreasing sequence {e(1), e(2), e(3), ...} of positive integers such that x = 1/e(1) + 1/(e(1)*e(2)) + 1/(e(1)*e(2)*e(3)) + .... The terms in the Engel expansion of x are obtained from the iterates of the map g(x) = x*(1 + floor(1/x)) - 1 by means of the formula e(n) = 1 + floor(1/g^(n-1)(x)). Here g^(n)(x) = g(g^(n-1)(x)) denotes the n-th iterate of g(x) with the convention g^(0)(x) = x.
In a similar way, the modified Engel expansion of x belonging to (0,1] is a sequence {E(1), E(2), E(3), ...} of positive integers such that x = 1/E(1) + 1/(E(1)*E(2)) + 1/(E(1)*E(2)*E(3)) + ... whose terms are obtained from the iterates of the harmonic sawtooth map h(x) = floor(1/x)*g(x). The general formula is E(1) = 1 + floor(1/x) and for n >= 1, E(n) = floor(1/h^(n-2)(x))*(1 + floor(1/h^(n-1)(x))). For further details see the Bala link.

Peter Bala, A modified Engel expansion for certain quadratic irrationals
Wikipedia, Engel Expansion

Cf. A002812, A028257, A220336 (p = 3), A220337 (p = 4), A220338 (p = 5).

Let x = sqrt(3) - 1. Then a(1) = ceiling(1/x) and for n >= 2, a(n) = floor(1/h^(n-2)(x))*ceiling(1/h^(n-1)(x)), where h^(n)(x) denotes the n-th iterate of the harmonic sawtooth map h(x), with the convention h^(0)(x) = x.
a(3*n+2) = 1/2*{2 + (2 + sqrt(3))^(2^n) + (2 - sqrt(3))^(2^n)} and
a(3*n+3) = (2 + sqrt(3))^(2^n) + (2 - sqrt(3))^(2^n), both for n >= 0.
For n >= 0, a(3*n+1) = 2. For n >= 1, a(3*n+2) = 2*(A002812(n-1))^2 and a(3*n+3) = 4*(A002812(n-1))^2 - 2.
Recurrence equations:
For n >= 1, a(3*n+2) = 2*{a(3*n-1)^2 - 2*a(3*n-1) + 1} and
a(3*n+3) = 2*a(3*n+2) - 2.
Put P(n) = Product_{k=1..n} a(k). Then we have the infinite Egyptian fraction representation sqrt(3) - 1 = Sum_{n>=1} 1/P(n) = 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*2) + ....

A220336 A modified Engel expansion for 4*sqrt(2) - 5.

Original entry on oeis.org

2, 4, 6, 2, 18, 34, 2, 578, 1154, 2, 665858, 1331714, 2, 886731088898, 1773462177794, 2, 1572584048032918633353218, 3145168096065837266706434, 2, 4946041176255201878775086487573351061418968498178, 9892082352510403757550172975146702122837936996354

Offset: 1

Peter Bala, Dec 12 2012

For a brief description of the modified Engel expansion of a real number see A220335.

Let p >= 2 be an integer and set Q(p) = (p - 1)*sqrt(p^2 - 1) - (p^2 - p - 1), so Q(3) = 4*sqrt(2) - 5. Iterating the identity Q(p) = 1/2 + 1/(2*(p+1)) + 1/(2*(p+1)*(2*p)) + 1/(2*(p+1)*(2*p))*Q(2*p^2-1) leads to a representation for Q(p) as an infinite series of unit fractions. The sequence of denominators of these unit fractions can be used to find the modified Engel expansion of Q(p). For further details see the Bala link. The present sequence is the case p = 3. For other cases see A220335 (p = 2), A220337 (p = 4) and A220338 (p = 5).

Peter Bala, A modified Engel expansion for certain quadratic irrationals
Wikipedia, Engel Expansion

Cf. A001601, A028257, A220335 (p = 2), A220337 (p = 4), A220338 (p = 5).

Define the map h(x) := floor(1/x)*(x*ceiling(1/x) - 1). Let x = 4*sqrt(2) - 5. Then a(1) = ceiling(1/x) and for n >= 2, a(n) = floor(1/h^(n-2)(x))*ceiling(1/h^(n-1)(x)), where h^(n)(x) denotes the n-th iterate of the map h(x), with the convention h^(0)(x) = x.
a(3*n+2) = 1/2*{2 + (1+sqrt(2))^(2^(n+1)) + (1-sqrt(2))^(2^(n+1))},
a(3*n+3) = {(1 + sqrt(2))^(2^(n+1)) + (1 - sqrt(2))^(2^(n+1))}, both
for n >= 0.
For n >= 0, a(3*n+1) = 2. For n >= 1, a(3*n+2) = 2*A001601(n)^2 and a(3*n+3) = 4*A001601(n)^2 - 2.
Recurrence equations:
For n >= 1, a(3*n+2) = 2*{a(3*n-1)^2 - 2*a(3*n-1) + 1} and
a(3*n+3) = 2*a(3*n+2) - 2.
Put P(n) = Product_{k=1..n} a(k). Then we have the infinite Egyptian fraction representation 4*sqrt(2) - 5 = Sum_{n>=1} 1/P(n) = 1/2 + 1/(2*4) + 1/(2*4*6) + 1/(2*4*6*2) + 1/(2*4*6*2*18) + ....

A054543 Engel series expansion (or "Egyptian product") for Catalan's constant G.

Original entry on oeis.org

2, 2, 2, 4, 4, 5, 5, 12, 13, 41, 110, 172, 248, 309, 3146, 5919, 21959, 22299, 30892, 401838, 1719239, 30576561, 262313756, 630913752, 3242181301, 3250783944, 13827502849, 40152067840, 137791590233, 2514510232695, 3217773878849

Offset: 1

Jeppe Stig Nielsen, Apr 09 2000

nonn

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 53-59.

Simon Plouffe, Table of n, a(n) for n = 1..212
Steven R. Finch, Catalan's Constant [Broken link]
Steven R. Finch, Catalan's Constant [From the Wayback machine]
Oleg Marichev, Jonathan Sondow, and Eric W. Weisstein, Catalan's Constant, MathWorld.
Eric Weisstein's World of Mathematics, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A006784, A028254, A028257, A006752, A104338, A014538, A153069, A153070, A118323.

EngelExp[A_,n_]:=Join[Array[1&,Floor[A]],First@Transpose@NestList[{Ceiling[1/Expand[ #[[1]]#[[2]]-1]],Expand[ #[[1]]#[[2]]-1]}&,{Ceiling[1/(A-Floor[A])],A-Floor[A]},n-1]]; EngelExp[N[Catalan,7! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

A028254 Engel expansion of sqrt(2).

Original entry on oeis.org

1, 3, 5, 5, 16, 18, 78, 102, 120, 144, 251, 363, 1402, 31169, 88630, 184655, 259252, 298770, 4196070, 38538874, 616984563, 1975413035, 5345718057, 27843871197, 54516286513, 334398528974, 445879679626, 495957494386, 2450869042061, 2629541150529, 4088114099885

Offset: 1

Naoki Sato (naoki(AT)math.toronto.edu)

nonn

For a number x (here sqrt(2)), define a(1) <= a(2) <= a(3) <= ... so that x = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... by x(1) = x, a(n) = ceiling(1/x(n)), x(n+1) = x(n)a(n) - 1.

			sqrt(2) = 1.4142135623730950488...
1 + 1/3 = 4/3 = 1.3333333333333333333...; sqrt(2) - 4/3 = 0.080880229...
1 + 1/3 + 1/15 = 7/5 = 1.4; sqrt(2) - 7/5 = 0.014213562373...
1 + 1/3 + 1/15 + 1/75 = 106/75 = 1.4133333333333333...; sqrt(2) - 106/75 = 0.000880229...

T. D. Noe, Table of n, a(n) for n = 1..300
Benoît Rittaud, La porte d’harmonie, Images des Mathématiques, CNRS, 2009 (in French).
Naoki Sato, Home page (broken link)
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Pythagoras's Constant

Cf. A002193 (decimal expansion), A006784 (for definition of Engel expansion), A028257 (Engel expansion of sqrt(3)).

expandEngel[A_, n_] := Join[Array[1 &, Floor[A]], First @ Transpose @ NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; expandEngel[N[2^(1/2), 7!], 47] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

More terms from Simon Plouffe, Jan 05 2002

A053977 Engel expansion of the Euler-Mascheroni constant gamma A001620 = 0.57721566... .

Original entry on oeis.org

2, 7, 13, 19, 85, 2601, 9602, 46268, 4812284, 147961485, 210810243, 814960948, 1003849128, 1016803038, 12917183059, 26242325020, 22215291139324, 30797877759859, 60139200644343, 121848657453426, 133555928335475

Offset: 1

Jeppe Stig Nielsen, Apr 02 2000

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

T. D. Noe, Table of n, a(n) for n=1..300
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant
Index entries for sequences related to Engel expansions

Cf. A006784, A028254, A028257.

EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ] ], Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]

More terms and additional comments from Mitch Harris, Jan 15 2001

A053980 Engel expansion of zeta(3) = 1.20206... .

Original entry on oeis.org

1, 5, 98, 127, 923, 5474, 16490, 25355, 37910, 85150, 1033216, 2290644, 7844861, 11170684, 18884358, 21481832, 35060787, 52399788, 201059261, 261533994, 9939708446, 211698940106, 3030068839686, 4326424644987, 6082687570463

Offset: 1

Jeppe Stig Nielsen, Apr 02 2000

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A006784, A028254, A028257.

Cf. A002117, A067912.

EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[[ 1 ] ]#[[ 2 ] ]-1 ] ], Expand[ #[[ 1 ] ]#[[ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]

More terms and additional comments from Mitch Harris, Jan 15 2001

A054544 Engel series expansion (or "Egyptian product") for Khintchine's constant.

Original entry on oeis.org

1, 1, 2, 3, 9, 70, 117, 503, 648, 1078, 12868, 41235, 178650, 377670, 394301, 546185, 2600672, 8729780, 41318679, 83367169, 525961060, 561571346, 1556964264, 1868773845, 15139200289, 27297789005, 30324107039, 56699922000

Offset: 0

Jeppe Stig Nielsen, Apr 09 2000

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 59-65.

Steven R. Finch, Khintchine's Constant [Broken link]
Steven R. Finch, Khintchine's Constant [From the Wayback machine]
Index entries for sequences related to Engel expansions

Cf. A002210, A006784, A028254, A028257.

cp's OEIS Frontend

A220335 A modified Engel expansion for sqrt(3) - 1.

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A220336 A modified Engel expansion for 4*sqrt(2) - 5.

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A054543 Engel series expansion (or "Egyptian product") for Catalan's constant G.

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A028254 Engel expansion of sqrt(2).

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A053977 Engel expansion of the Euler-Mascheroni constant gamma A001620 = 0.57721566... .

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A053980 Engel expansion of zeta(3) = 1.20206... .

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Mathematica

Extensions

A054544 Engel series expansion (or "Egyptian product") for Khintchine's constant.

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