A028254 -id:A028254 - cp's OEIS frontend

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

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 *)

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.

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

A091831 Pierce expansion of 1/sqrt(2).

Original entry on oeis.org

1, 3, 8, 33, 35, 39201, 39203, 60245508192801, 60245508192803, 218662352649181293830957829984632156775201, 218662352649181293830957829984632156775203

Offset: 0

Benoit Cloitre, Mar 09 2004

nonn

If u(0)=exp(1/m) m integer>1 and u(n+1)=u(n)/frac(u(n)) then floor(u(n))=m*n.

G. C. Greubel, Table of n, a(n) for n = 0..14
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, On a problem of Alfred Renyi, Economics Working Paper No. 340.
Eric Weisstein's World of Mathematics, Pierce Expansion

Cf. A006275, A006276, A006283.

Cf. A006784 (Pierce expansion definition), A028254

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[2^(-1/2), 7!], 17] (* G. C. Greubel, Nov 13 2016 *)

r=sqrt(2);for(n=1,10,r=r/(r-floor(r));print1(floor(r),","))

Let u(0)=sqrt(2) and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n)).
1/sqrt(2)= 1/a(1) - 1/a(1)/a(2) + 1/a(1)/a(2)/a(3) - 1/a(1)/a(2)/a(3)/a(4)...
limit n -> infinity a(n)^(1/n) = e.

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

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

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A220397 A modified 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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A091831 Pierce expansion of 1/sqrt(2).

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

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