A006784 -id:A006784 - cp's OEIS frontend

A067919 Engel expansion of sin(1).

2, 2, 3, 11, 14, 27, 28, 66, 212, 231, 552, 2842, 3774, 6038, 6784, 10950, 32948, 78591, 97875, 98342, 123569, 139837, 159698, 1102838, 3256476, 20329622, 34385124, 60999878, 82669919, 85820365, 389915995, 4274338879, 18907353107, 62875944378, 74931184173

Offset: 1

Benoit Cloitre, Mar 03 2002

			sin(1) = 0.84147... = A049469 has the Engel expansion 1/2 + 1/(2*2) + 1/(2*2*3) + ...

See A006784 for explanation of Engel expansions.

Cf. A049469, A084651, A084652.

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[Sin[1],6! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2009 *)

s=sin(1); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

a(1) inserted by Hauke Worpel (hw1(AT)email.com), Jun 01 2003

Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

A068379 Engel expansion of sinh(1/2).

Original entry on oeis.org

2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224, 28560

Offset: 1

Benoit Cloitre, Mar 03 2002

Cf. A006784 for Engel expansion definition.

The MathWorld link mentions the closed form of the Engel expansion of sinh(1) = A068377. - Georg Fischer, Nov 22 2020

			sinh(1/2) = 1/2 + 1/(2*24) + 1/(2*24*80) + 1/(2*24*80*168) + 1/(2*24*80*168*288) + ... = 0.52109530549374736162242562641... = A334367.

Vincenzo Librandi, Table of n, a(n) for n = 1..10001
Eric W. Weisstein's World of Mathematics, Engel Expansion.
Wikipedia, Engel Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002943, A006784, A033586, A068377, A068380, A334367.

[2] cat [8*(n*(2*n-3)+1): n in [2..50]]; // Vincenzo Librandi, Jan 31 2012

Table[If[n==1, 2, 8*(n*(2*n-3)+1)], {n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{3,-3,1},{2,24,80,168},50] (* Harvey P. Dale, Mar 21 2017 *)

a(n)=if(n<=1,2,8*(n*(2*n-3)+1)) \\ Charles R Greathouse IV, Jan 31 2012

a(n) = 8*(n*(2*n-3)+1) for n > 1, a(1)=2.
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 2*x*(1+9*x+7*x^2-x^3)/(1-x)^3. (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = (3-log(2))/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/4 - Pi/16 - log(2)/8. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: 2*(4*exp(x)*(1 - x + 2*x^2) + (x - 4)).
a(n) = 2*A033586(n-1) for n >= 2.
a(n) = 4*A002943(n-1) for n >= 2. (End)

Edited, offset 1 and a(1)=2 in programs and b-file by Georg Fischer, Nov 22 2020

A068380 Engel expansion of sinh(1/3).

Original entry on oeis.org

3, 54, 180, 378, 648, 990, 1404, 1890, 2448, 3078, 3780, 4554, 5400, 6318, 7308, 8370, 9504, 10710, 11988, 13338, 14760, 16254, 17820, 19458, 21168, 22950, 24804, 26730, 28728, 30798, 32940, 35154, 37440, 39798, 42228, 44730, 47304, 49950, 52668, 55458, 58320, 61254

Offset: 1

Benoit Cloitre, Mar 03 2002

Cf. A006784 for the definition of the Engel expansion.

The MathWorld link mentions the closed form of the Engel expansion of sinh(1). - Georg Fischer, Nov 22 2020

			sinh(1/3) = 1/3 + 1/(3*54) + 1/(3*54*180) + 1/(3*54*180*378) + 1/(3*54*180*378*648) + ... = 0.33954055725615013910126061...

Eric Weisstein's World of Mathematics, Engel Expansion.
Wikipedia, Engel Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784, A002943, A068377, A068379.

LinearRecurrence[{3, -3, 1}, {3, 54, 180, 378}, 50]

PARI
```
a(n)=if(n<=1, 3, 18*(n*(2*n-3)+1));
```

my(x='x+O('x^43)); Vec(3*x*(1+15*x+9*x^2-x^3)/(1-x)^3) \\ Elmo R. Oliveira, May 29 2025

a(n) = 18*(n*(2*n-3)+1) for n > 1, a(1)=3. - Ralf Stephan, Sep 03 2003
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 3*x*(x^3-9*x^2-15*x-1)/(x-1)^3. (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = (4-log(2))/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4/9 - Pi/36 - log(2)/18. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: 3*(6*exp(x)*(1 - x + 2*x^2) + (x - 6)).
a(n) = 9*A002943(n-1) for n >= 2. (End)

Edited, offset 1 and a(1)=3 by Georg Fischer, Nov 23 2020

More terms from Elmo R. Oliveira, May 29 2025

A225208 Engel expansion of the positive root of x^x^x^x = 2.

Original entry on oeis.org

1, 3, 3, 52, 106, 260, 279, 334, 491, 536, 728, 1161, 5678, 15183, 41437, 189034, 281965, 1118629, 3473978, 32869874, 82525851, 159312757, 424570638, 472381891, 563118608, 579529452, 1426303902, 2330077798, 2991863700, 25850322702, 34547004920, 37294688664

Offset: 1

Alois P. Heinz, May 01 2013

nonn

It is not known if the positive root of x^x^x^x = 2 is a rational number and, in consequence, whether this sequence is finite or not.

			1.44660143242986417459733398759766148...

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

Alois P. Heinz, Table of n, a(n) for n = 1..100
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
Wikipedia, Engel Expansion
Wikipedia, Tetration, Open questions
Index entries for sequences related to Engel expansions

Cf. A225134 (decimal expansion), A225153 (continued fraction).

Digits:= 500:
c:= solve(x^(x^(x^x))=2, x):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r),
        engel(r*ceil(1/r)-1, n-1)][]):
engel(evalf(c), 39);

A248917 a(n) = 2^n * n^2 + 1.

Original entry on oeis.org

1, 3, 17, 73, 257, 801, 2305, 6273, 16385, 41473, 102401, 247809, 589825, 1384449, 3211265, 7372801, 16777217, 37879809, 84934657, 189267969, 419430401, 924844033, 2030043137, 4437573633, 9663676417, 20971520001, 45365592065, 97844723713, 210453397505, 451508436993

Offset: 0

Paul Curtz, Oct 22 2014

Binomial transform of A118239 (Engel expansion of cosh(1)).
Table of successive differences of a(n):
1,   3,   17,  73, 257, 801, 2305,...
2,   14,  56, 184, 544, 1504,...
12,  42, 128, 360, 960,...
30,  86, 232, 600,...
56, 146, 368,...
90, 222,...
132,...
etc.
Via b(n) = 0, 0, 0 followed by A055580(n), i.e., 0, 0, 0, 1, 7, 31, 111, ... (the main sequence for the recurrence), a link can be found between a(n) and A002064(n): c(n) = b(n+1) - 2*b(n) = 0, 0, 1, 5, 17, 49, 129, ... (the main sequence for the signature (5, -8, 4)).

			a(3) = 9 * 8 + 1 = 73.
a(4) = 16 * 16 + 1 = 257.
a(5) = 25 * 32 + 1 = 801.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A000225, A002064 (Cullen numbers), A006784, A007758, A055580, A118239, A168298.

[2^n*n^2+1: n in [0..30]]; // Vincenzo Librandi, Oct 29 2016

Table[n^2 * 2^n + 1, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *)
LinearRecurrence[{7,-18,20,-8}, {1,3,17,73}, 25] (* G. C. Greubel, Oct 28 2016 *)

Vec(-(12*x^3-14*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Oct 22 2014

a(n)=n^2<Charles R Greathouse IV, Oct 22 2014

a(n) = 4*a(n-1) - 4*a(n-2) + 2^(n+1) + 1.
a(n) = A007758(n) + 1.
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4). - Jean-François Alcover, Oct 22 2014
G.f.: -(12*x^3-14*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Oct 22 2014
E.g.f.: exp(x) + 2*x*(1 + 2*x)*exp(2*x). - G. C. Greubel, Oct 28 2016

A059176 Engel expansion of sqrt(5) = 2.23606...

Original entry on oeis.org

1, 1, 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

Mitch Harris

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.

Simon Plouffe, Table of n, a(n) for n = 1..883
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. A002163.

Essentially the same as A028259.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Sqrt[5], 7!], 50] (* modified by G. C. Greubel, Dec 26 2016 *)

A059194 Engel expansion of 1/e^2 = 0.135335... .

Original entry on oeis.org

8, 13, 14, 21, 87, 92, 119, 444, 472, 473, 548, 5380, 7995, 100393, 589494, 2034930, 12322338, 21633910, 55986423, 164342975, 6502609245, 22562439736, 26621735244, 39286977900, 576511092268, 892451075829, 1050206120774, 2228669763793, 3336969029043

Offset: 1

Mitch Harris

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 and T. D. Noe, Table of n, a(n) for n = 1..1000 [Terms 1 to 300 computed by T. D. Noe; Terms 301 to 1000 computed by G. C. Greubel, Dec 28 2016]
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. A092553.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[1/E^2, 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A059195 Engel expansion of log(Pi) = 1.14473... .

Original entry on oeis.org

1, 7, 77, 107, 150, 167, 7091, 27852, 31790, 34069, 327724, 416403, 4669290, 20206510, 2218014305, 4524826037, 4576058224, 5496581959, 15869888136, 91151928112, 104430320239, 202761572952, 218933128153, 937032410920, 1044739832405, 15262515810234

Offset: 1

Mitch Harris

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. A053510.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Log[Pi], 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A059200 Engel expansion of -log(log(2)) = 0.36651292... .

Original entry on oeis.org

3, 11, 11, 23, 62, 66, 466, 1450, 7617, 95677, 100963, 153329, 966054, 4744661, 23899231, 25086529, 52363821, 100389201, 201892089, 261170111, 312778184, 527002514, 1235004065, 1623652949, 2309078745, 8274570969

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner 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. A074785.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[-Log[Log[2]], 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A060787 a(n) = 18(n - 2)(2*n - 5).

Original entry on oeis.org

0, 18, 108, 270, 504, 810, 1188, 1638, 2160, 2754, 3420, 4158, 4968, 5850, 6804, 7830, 8928, 10098, 11340, 12654, 14040, 15498, 17028, 18630, 20304, 22050, 23868, 25758, 27720, 29754, 31860, 34038, 36288, 38610, 41004, 43470, 46008, 48618, 51300, 54054, 56880, 59778

Offset: 2

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 28 2001

Except for first term Engel expansion of cosh(1/3); cf. A006784 for Engel expansion definition. - Benoit Cloitre, Mar 03 2002

Luigi Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906. p. 341.
Henri Brocard and Timoléon Lemoyne, Courbes géométriques remarquables (courbes spéciales) Planes et Gauches. Tome I, Paris: Albert Blanchard, 1967 [First publ. 1919]; see p. 135.

Harry J. Smith, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784.

a[n_] := 18*(n-2)*(2*n-5); Array[a, 50, 2] (* Amiram Eldar, May 05 2025 *)

a(n) = 18*(n - 2)*(2*n - 5) \\ Harry J. Smith, Jul 11 2009

G.f.: 18*x^3*(1 + 3*x)/(1 - x)^3. - Colin Barker, Feb 29 2012
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=3} 1/a(n) = log(2)/9.
Sum_{n>=3} (-1)^(n+1)/a(n) = Pi/36 - log(2)/18. (End)

cp's OEIS Frontend

A067919 Engel expansion of sin(1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A068379 Engel expansion of sinh(1/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A068380 Engel expansion of sinh(1/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A225208 Engel expansion of the positive root of x^x^x^x = 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

A248917 a(n) = 2^n * n^2 + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A059176 Engel expansion of sqrt(5) = 2.23606...

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A059194 Engel expansion of 1/e^2 = 0.135335... .

Views

Author

Keywords

Comments

References

A060787 a(n) = 18(n - 2)(2*n - 5).