A220398 -id:A220398 - cp's OEIS frontend

A220393 A modified Engel expansion of Pi.

1, 1, 1, 8, 14, 2, 2, 3, 4, 5, 96, 115, 8, 2, 2, 2, 81, 160, 2, 6, 355, 140, 4, 12, 6, 2, 2, 3, 4, 3, 46, 66, 4, 2, 9, 16, 3, 4, 3, 4, 2, 2, 4, 9, 4, 2, 4, 33, 20, 2, 3, 4, 2, 2

Offset: 1

Peter Bala, Dec 13 2012

The Engel expansion of a positive real number x 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 mapping g(x) := x*(1 + floor(1/x)) - 1 by means of the formula e(n) = 1 + floor(1/g^(n)(x)).
Let h(x) = floor(1/x)*g(x). This is called the harmonic sawtooth map by Crowley. Then the modified Engel expansion of a real number 0 < x <= 1 is a sequence {a(1), a(2), a(3), ...} of positive integers such that x = 1/a(1) + 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) + ... whose terms are obtained from the iterates of the harmonic sawtooth map h(x) by the formulas a(1) = 1 + floor(1/x) and, for n >= 2, a(n) = floor(1/h^(n-2)(x))*{1 + floor(1/h^(n-1)(x))}. Here h^(n)(x) = h(h^(n-1)(x)) denotes the n-th iterate of the map h(x), with the convention that h^(0)(x) = x. For further details see the Bala link.
When the real number x > 1 with, say, floor(x) = m, the modified Engel expansion of x is found by first calculating the modified Engel expansion of x - m and then prepending a sequence of m 1's to this.

Peter Bala, A modified Engel expansion
Wikipedia, Engel Expansion

Cf. A006784, A220335, A220336, A220337, A220338, A220394, A220395, A220396, A220397, A220398.

Let x = Pi - 3. Then a(1) = a(2) = a(3) = 1, a(4) = ceiling(1/x) and, for n >= 1, a(n+4) = floor(1/h^(n-1)(x))*ceiling(1/h^(n)(x)).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion Pi = Sum_{n>=1} 1/P(n) = 1/1 + 1/1 + 1/1 + 1/8 + 1/(8*14) + 1/(8*14*2) + 1/(8*14*2*2) + .... For n >= 4, the error made in truncating this series to n terms is less than the n-th term.

A028259 Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .

Original entry on oeis.org

1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, 177, 263, 332, 389, 4102, 4575, 5081, 9962, 18316, 86613, 233239, 342534, 964372, 1452850, 7037119, 7339713, 8270361, 12855437, 15900982, 19211148, 1365302354, 1565752087, 1731612283

Offset: 1

Naoki Sato (naoki(AT)math.toronto.edu), Dec 11 1999

Cf. A006784 for definition of Engel expansion.
(sqrt(5) - 1)/2 = 1/(golden ratio) has the same Engel expansion starting with a(2). - G. C. Greubel, Oct 16 2016
Each Engel expansion for a constant x (x > 0) starts with floor(x) 1's. After that, the mean growth of the terms is by a factor e for most constants x, i.e., the order of magnitude of the n-th entry is exp(n-floor(c)), for n large enough. This comment is similar to the fact that for the continued fraction terms of most constants, the geometric mean of those terms equals the Khintchine constant for n large enough. Moreover, note that for the golden section, all continued fraction terms are 1 and thus do not obey the Gauss-Kuzmin distribution which leads to the Khintchine constant (i.e., the Khintchine measure for the golden ratio is 1), but the Engel expansion does obey the statistic behavior for most constants. - A.H.M. Smeets, Aug 24 2018

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

A.H.M. Smeets, Table of n, a(n) for n = 1..2406 (Terms 1 through 300 from T. D. Noe, 301 through 698 from Simon Plouffe, and 699 through 1500 from G. C. Greubel)
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, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Golden Ratio
Index entries for sequences related to Engel expansions

Cf. A001622, A220398.

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 ] ]

j = 0
while j<3100000:
# to obtain n correct Engel expansion terms about n^2/2 continued fraction steps are needed; 3100000 is a safe bound
    if j == 0:
        p0, q0 = 1, 1
    elif j == 1:
        p1, q1 = p0+1, 1
    else:
        p0, p1 = p1, p1+p0
        q0, q1 = q1, q1+q0
    j = j+1
i = 0
while i < 2410:
    i = i+1
    a = q0//p0+1
    print(i, a)
    p0 = a*p0-q0
# A.H.M. Smeets, Aug 24 2018

Limit_{n -> oo} log(a(n + floor(golden ratio)))/n = 1.

Corrected and extended by Vladeta Jovovic, Aug 16 2001

A220394 A modified Engel expansion of exp(1).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 2, 10, 99, 20, 2, 2, 2, 2, 2, 2, 3, 6, 4, 8, 14, 2, 2, 4, 6, 10, 252, 81, 30, 28, 31, 60, 4, 6, 3, 4, 2, 2, 2, 2, 19, 54, 8, 6, 22, 63, 4, 2, 4, 6, 2, 2, 5, 12, 4, 2, 2, 2, 2, 6, 15, 10, 348, 172, 2, 2, 4, 6, 4, 30, 207, 220

Offset: 1

Peter Bala, Dec 13 2012

See A220393 for a description of the modified Engel expansion of a positive real number. For further details see the Bala link.

The Engel expansion for exp(1) is the sequence of positive integers A000027.

Peter Bala, A modified Engel expansion
Wikipedia, Engel Expansion

Cf. A000027, A220335, A220336, A220337, A220338, A220393, A220395, A220396, A220397, A220398.

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x) where x = exp(1) - 2, then a(1) = a(2) = 1, a(3) = ceiling(1/x) and, for n >= 1, a(n+3) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion exp(1) = Sum_{n>=1} 1/P(n) = 1/1 + 1/1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*5) + 1/(2*3*4*5*8) + .... For n >= 3, the error made in truncating this series to n terms is less than the n-th term.

A220395 A modified Engel expansion of log(2).

Original entry on oeis.org

2, 3, 8, 6, 2, 4, 93, 60, 2, 2, 2, 2, 3, 12, 10, 2, 2, 14, 52, 6, 5, 8, 2, 2, 5, 8, 2, 2, 3, 4, 14, 273, 40, 2, 3, 4, 4, 12, 27, 16, 14, 26, 4, 6, 4, 6, 2, 3, 12, 10, 4, 6, 14, 65, 12, 8, 6, 2, 7, 90, 294, 40, 2, 2, 32, 155, 8, 7, 12, 2, 2, 2, 2, 4, 6, 3, 10

Offset: 1

Peter Bala, Dec 13 2012

See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

Peter Bala, A modified Engel expansion
Wikipedia, Engel Expansion

Cf. A059180, A220335, A220336, A220337, A220338, A220393, A220394, A220396, A220397, A220398.

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = log(2). Then a(1) = 1 + floor(1/x) and, for n >= 1, a(n+1) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion log(2) = Sum_{n>=1} 1/P(n) = 1/2 + 1/(2*3) + 1/(2*3*8) + 1/(2*3*8*6) + 1/(2*3*8*6*2) + .... The error made in truncating this series to n terms is less than the n-th term.

A220396 A modified Engel expansion of the Euler-Mascheroni constant gamma.

Original entry on oeis.org

2, 7, 18, 4, 2, 2, 3, 1466, 1464, 9, 24, 4, 2, 9, 104, 60, 8, 2, 3, 6, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 32, 30, 2, 13, 36, 6, 4, 3, 6, 6, 4, 4, 6, 2, 4, 6, 2, 4, 6, 9, 24, 4, 5, 8, 2, 2, 2, 2, 2, 3, 20

Offset: 1

Peter Bala, Dec 13 2012

See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

Peter Bala, A modified Engel expansion
Wikipedia, Engel Expansion

Cf. A001620, A053977, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220397, A220398.

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = gamma (see A001620). Then a(1) = 1 + floor(1/x) and, for n >= 1, a(n+1) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion sqrt(2) = Sum_{n>=1} 1/P(n) = 1/2 + 1/(2*7) + 1/(2*7*18) + 1/(2*7*18*4) + 1/(2*7*18*4*2) + .... The error made in truncating this series to n terms is less than the n-th term.

A220397 A modified Engel expansion of sqrt(2).

Original entry on oeis.org

1, 3, 6, 4, 2, 2, 4, 6, 23, 66, 108, 7738, 290, 9, 24, 32, 30, 4, 6, 3, 6, 24, 22, 2, 6, 20, 6, 9, 16, 5, 12, 4, 12, 22, 5, 8, 3, 6, 4, 2, 2, 4, 6, 2, 2, 2, 2, 13, 24, 2, 3, 4, 2, 2, 2, 2, 23, 44, 21, 40, 8, 14, 3, 6, 12, 10, 11, 30, 4, 4, 9, 4, 3, 4, 2, 16, 45, 46, 528

Offset: 1

Peter Bala, Dec 13 2012

See A220393 for a definition of the modified Engel expansion of a positive real number. For further details see the Bala link.

Peter Bala, A modified Engel expansion
Wikipedia, Engel Expansion

Cf. A028254, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220398.

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = sqrt(2) - 1. Then a(1) = 1, a(2) = ceiling(1/x) and, for n >= 1, a(n+2) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion sqrt(2) = Sum_{n>=1} 1/P(n) = 1 + 1/3 + 1/(3*6) + 1/(3*6*4) + 1/(3*6*4*2) + 1/(3*6*4*2*2) + .... For n >= 2, the error made in truncating this series to n terms is less than the n-th term.

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A220393 A modified Engel expansion of Pi.

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A028259 Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .

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A220394 A modified Engel expansion of exp(1).

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A220395 A modified Engel expansion of log(2).

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A220396 A modified Engel expansion of the Euler-Mascheroni constant gamma.

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A220397 A modified Engel expansion of sqrt(2).

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