A006784 -id:A006784 - cp's OEIS frontend

A059193 Engel expansion of 1/e = 0.367879... .

3, 10, 28, 54, 88, 130, 180, 238, 304, 378, 460, 550, 648, 754, 868, 990, 1120, 1258, 1404, 1558, 1720, 1890, 2068, 2254, 2448, 2650, 2860, 3078, 3304, 3538, 3780, 4030, 4288, 4554, 4828, 5110, 5400, 5698, 6004, 6318, 6640, 6970, 7308, 7654, 8008, 8370, 8740

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

Friedrich 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 27 2016]
Friedrich 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.
Paul 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.
Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.
Eric Weisstein's World of Mathematics, Engel Expansion.
Index entries for sequences related to Engel expansions.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784, A068985.

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, 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)
Join[{3}, Table[2*(2*n+1)*(n-1), {n, 1, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Join[{3},LinearRecurrence[{3,-3,1},{10,28,54},50]] (* Harvey P. Dale, May 10 2012 *)

Vec(x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3 + O(x^50)) \\ G. C. Greubel, Dec 27 2016

a(n) = 2*(2*n+1)*(n-1) (for n>1) follows from 1/e = Sum_{n>=1} (1/(2*n)! - 1/(2*n+1)!). - Helena Verrill (verrill(AT)math.lsu.edu), Jan 19 2004
a(1)=3, a(2)=10, a(1)=28, a(2)=54, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 10 2012
From G. C. Greubel, Dec 27 2016: (Start)
G.f.: x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3.
E.g.f.: 2 + 3*x + 2*(2*x^2 + x - 1)*exp(x). (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = 7/9 - log(2)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/9 + Pi/12 - log(2)/6. (End)

A067912 Engel expansion of zeta(4) = Pi^4/90 = Sum_{i>0} 1/i^4.

Original entry on oeis.org

1, 13, 15, 19, 132, 1474, 1977, 10392, 12992, 44777, 59412, 170685, 217607, 704791, 818133, 1387423, 2208674, 3206215, 12732462, 13962681, 24593168, 39744274, 55804517, 130269696, 426536424, 546807194, 1030799587, 1139987135

Offset: 1

Benoit Cloitre, Mar 03 2002

G. C. Greubel, Table of n, a(n) for n = 1..1000

See A006784 for explanation of Engel expansions.

Cf. A059186, A053980.

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[Pi^4/90, 7!], 20] (* G. C. Greubel, Dec 26 2016 *)

A118239 Engel expansion of cosh(1).

Original entry on oeis.org

1, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372

Offset: 1

Eric W. Weisstein, Apr 17 2006

Differs from A002939 only in first term.

This sequence is also the Pierce expansion of cos(1). - G. C. Greubel, Nov 14 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Engel Expansion.
Eric Weisstein's World of Mathematics, Pierce Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784, A002939, A068377.

Join[{1}, Table[(2 n - 2) (2 n - 3), {n, 2, 50}]] (* Bruno Berselli, Aug 04 2015 *)
Join[{1}, LinearRecurrence[{3,-3,1},{2,12,30},25]] (* G. C. Greubel, Oct 27 2016 *)
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[Cos[1] , 7!], 50] (* G. C. Greubel, Nov 14 2016 *)

a(n)=max(4*n^2-10*n+6, 1) \\ Charles R Greathouse IV, Oct 22 2014

A118239 = lambda n: falling_factorial(n*2,2) if n>0 else 1
print([A118239(n) for n in (0..46)]) # Peter Luschny, Aug 04 2015

a(n) = A002939(n-1) = 2*(n-1)*(2*n-3) for n>1.
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1 - x + 9*x^2 - x^3)/(1-x)^3. (End)
E.g.f.: -6 + x + 2*(3 - 3*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 27 2016
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = log(2) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - Pi/4 + log(2)/2. (End)

A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

Original entry on oeis.org

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

Offset: 1

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

			2.8702221569733963308194588658111996012403192622809957012...

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

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.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A006784, A064648, A101689, A001405, A070910, A096789, A141827.

evalf(BesselI(0, 2) + BesselI(1, 2) - 1, 100); # Peter Bala, Jul 02 2016

First@ RealDigits@ N[Sum[1/Product[Ceiling[r/2], {r, n}], {n, 1000}], 100] (* Original program amended to generate output by Michael De Vlieger, Jul 03 2016 *)
RealDigits[3 - HypergeometricPFQ[{1, 1}, {3, 3, 3}, 1]/8, 10, 100][[1]] (* Vaclav Kotesovec, Jul 03 2016 *)

From Peter Bala, Jul 01 2016: (Start)
Constant c = 1/1 + 1/(1*1) + 1/(1*1*2) + 1/(1*1*2*2) + 1/(1*1*2*2*3) + 1/(1*1*2*2*3*3) + ... = Sum_{n >= 1} binomial(n,floor(n/2))/n!.
Alternative series representations:
c = 3 - Sum_{n >= 2} 1/(n*(n - 1)*n!^2);
c = 1 + Sum_{n >= 1} (n + 2)/(n!*(n + 1)!);
c = 5/3 + 1/3*Sum_{n >= 2} (n + 1)*(n + 2)/n!^2;
c = A070910 + A096789 - 1.
Continued fraction: c = 3 - 1/(8 - 4/(14 - 9/(32 - ... - (n-1)^2/(n^2 + n + 2 - ...)))). See comments in A141827. (End)

A182503 Engel expansion of the Dottie number, A003957.

Original entry on oeis.org

2, 3, 3, 4, 5, 15, 17, 66, 196, 233, 284, 375, 1613, 2131, 3574, 14122, 24171, 49097, 56871, 69361, 193406, 243145, 289951, 682749, 14501588, 21191773, 121635191, 810759781, 1292785381, 136110231377, 294401497761

Offset: 1

Ben Branman, May 02 2012

Dottie number = 1/2 + 1/2/3 + 1/2/3/3 + 1/2/3/3/4 + 1/2/3/3/4/5 + 1/2/3/3/4/5/15 +...

G. C. Greubel, Table of n, a(n) for n = 1..500
Jean-Christophe Pain, An exact series expansion for the Dottie number, arXiv:2303.17962 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Dottie Number
Index entries for sequences related to Engel expansions

Cf. A003957, A006784.

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]]; z = FindRoot[x == Cos[x], {x, 1}, WorkingPrecision -> 10000][[1, -1]]; EngelExp[z, 30]

A014012 Engel expansion of 1/Pi.

Original entry on oeis.org

4, 4, 11, 45, 70, 1111, 4423, 5478, 49340, 94388, 200677, 308749, 708066, 711391, 1113024, 4342375, 4529119, 8061070, 12060867, 56215509, 69737317, 124001030, 214920537, 471564389, 891380746, 4293367334, 5031151602, 9832878719, 15034446439, 15481444638

Offset: 1

Simon Plouffe

nonn

Simon Plouffe, Table of n, a(n) for n = 1..973 [Terms 1 through 300 were computed by T. D. Noe]
Index entries for sequences related to Engel expansions

See A006784 for definition.

a(n):=proc(s)
local
i, j, max, aa, bb, lll, prod, S, T, kk;
    S := evalf(abs(s));
    max := 10^(Digits - 10);
    prod := 1;
    lll := [];
    while prod <= max do
        T := 1 + trunc(1/S);
        S := frac(S*T);
        lll := [op(lll), T];
        prod := prod*T
    end do;
    RETURN(lll)
end;
### Enter a real number and the program will return the Engel expansion of that number, the number of terms is adjusted to the output
# Simon Plouffe, Apr 23 2016

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[1/Pi,7! ],50] (* Vladimir Joseph Stephan Orlovsky, Jun 08 2009 *)

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

A059180 Engel expansion of log(2).

Original entry on oeis.org

2, 3, 7, 9, 104, 510, 1413, 2386, 40447, 87110, 124975, 1565154, 1766158, 2440919, 2637001, 9192874, 24998746, 73973182, 88828340, 432049320, 470421590, 477600793, 3313014448, 4571423959, 28839435286, 40818751774

Offset: 1

Mitch Harris

See A006784 for the 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 27 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.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Natural Logarithm of 2
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. A002162 (decimal expansion of the natural logarithm of 2).

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[2], 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

A059186 Engel expansion of Pi^2/6, or zeta(2) = 1.64493.

Original entry on oeis.org

1, 2, 4, 7, 9, 22, 35, 79, 2992, 3597, 17523, 28632, 41470, 53093, 57406, 14504930, 42622213, 188335162, 322429556, 1023003875, 1328535963, 3138645732, 11618168524, 137721814936, 156929353744, 166732460513, 813398686532

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 27 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. A013661, A053980, A067912.

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[Pi^2/6, 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

cp's OEIS Frontend

A059193 Engel expansion of 1/e = 0.367879... .

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A067912 Engel expansion of zeta(4) = Pi^4/90 = Sum_{i>0} 1/i^4.

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A118239 Engel expansion of cosh(1).

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A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

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A182503 Engel expansion of the Dottie number, A003957.

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A014012 Engel expansion of 1/Pi.

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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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