A006784 -id:A006784 - cp's OEIS frontend

A067239 a(0)=1, a(n) = 8n(2*n-1).

1, 8, 48, 120, 224, 360, 528, 728, 960, 1224, 1520, 1848, 2208, 2600, 3024, 3480, 3968, 4488, 5040, 5624, 6240, 6888, 7568, 8280, 9024, 9800, 10608, 11448, 12320, 13224, 14160, 15128, 16128, 17160, 18224, 19320, 20448, 21608, 22800, 24024

Offset: 0

Benoit Cloitre, Mar 03 2002

Engel expansion of cosh(1/2).

Also, 8 times hexagonal numbers (8*A000384(n) = A152750(n)), for n > 0. - Omar E. Pol, Dec 14 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784 for Engel expansion definition.

Cf. A000384, A152750.

s=0;lst={s};Do[s+=n++ +8;AppendTo[lst, s], {n, 0, 8!, 32}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

Vec((1+5*x+27*x^2-x^3)/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Apr 13 2012

a(n) = a(n-1) + 32*n - 24 (with a(1)=8). - Vincenzo Librandi, Dec 15 2010
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
G.f.: (1 + 5*x + 27*x^2 - x^3)/(1-x)^3. (End)
E.g.f.: 1 + 8*exp(x)*x*(1 + 2*x). - Elmo R. Oliveira, Dec 15 2024
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=0} 1/a(n) = log(2)/4 + 1.
Sum_{n>=0} (-1)^n/a(n) = 1 - Pi/16 + log(2)/8. (End)

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.

A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

Original entry on oeis.org

1, 4, 4, 4, 4, 6, 11, 11, 11, 14, 61, 266, 1006, 1030, 1261, 6264, 7583, 7979, 7986, 12386, 80041, 87434, 130927, 270073, 1653819, 1715177, 1973657, 3483485, 12346987, 17531499, 21237674, 84103203, 195088616, 725688944, 2813572082, 3138084145, 10870485195

Offset: 1

Alois P. Heinz, Apr 23 2009

Cf. A006784 for definition of Engel expansion.

			hz = 1.3327473824328992250086010983738997044167439822598445365797 ...

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

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
Index entries for sequences related to Engel expansions

Cf. A158468 (decimal expansion), A158469 (continued fraction).

hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity):
engel:= (r,n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
Digits:=300:
engel(evalf(hz), 39);

Some terms corrected by Alois P. Heinz, Nov 22 2020

A199880 Engel expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.

Original entry on oeis.org

3, 4, 8, 12, 15, 33, 70, 4338, 22062, 46566, 98091, 255284, 2715877, 10855925, 150153128, 10009347774, 34679420772, 43644678207, 74587800101, 229110893125, 233558717156, 286861037311, 299617642336, 312870987050, 1632483095154, 31761226898013, 66327161231576

Offset: 1

Alois P. Heinz, Nov 11 2011

nonn

Cf. A006784 for definition of Engel expansion.

			0.42801103796472992390204...

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

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner 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
Index entries for sequences related to Engel expansions

Cf. A199814 (decimal expansion), A199879 (continued fraction).

f:= x-> (x^(x^x))^(x^x): g:= x-> x^(x^((x^x)^x)):
Digits:= 700:
xv:= fsolve(f(x)=g(x), x=0..0.99):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
engel(xv, 39);

A222482 Engel expansion of cos(1)/(1+cos(1)).

Original entry on oeis.org

3, 20, 22, 40, 68, 248, 7163, 28663, 50059, 64574, 638169, 761733, 2537764, 2925739, 3363073, 4977902, 5646039, 57212854, 159650555, 219684539, 453524713, 459239955, 2002180165, 3234082460, 14965375439, 50298730245, 89316768769, 464076054936, 520232391320

Offset: 1

Alois P. Heinz, Feb 21 2013

nonn

Cf. A006784 for definition of Engel expansion.

			0.35077679479523758155811675...

Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A222480 (decimal expansion), A222481 (continued fraction), A049470 (cos(1)), A073448 (1/cos(1)).

Digits:= 1000:
b:= evalf(1/(1+1/cos(1))):  engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
engel(b, 39);

cos(1)/(1+cos(1)) = 1/(1+1/cos(1)) = 1/(1+sec(1)).

A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 16 2013

Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - Robert G. Wilson v, Jun 27 2015
Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...
Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.
Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).
Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).
Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.
e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.
1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....

			2.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Engel Expansion
Googology Wiki, Double factorial

Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).

SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Apr 01 2019

RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)

default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ G. C. Greubel, Apr 01 2019

numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Apr 01 2019

A329912 Engel expansion of exp(Pi/4).

Original entry on oeis.org

1, 1, 6, 7, 9, 17, 57, 283, 326, 791, 10332, 17303, 24977, 85451, 96025, 192273, 337177, 700071, 1394732, 2514757, 73904827, 176943055, 340834596, 663816066, 833303392, 2045234708, 2352089677, 7248164506, 10106625539, 32495772149, 54837573240, 60139816999

Offset: 1

Alois P. Heinz, Nov 23 2019

nonn

Cf. A006784 for definition of Engel expansion.

Greg Egan, Puzzle in which this value arises naturally
F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner 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.
Grant Sanderson and Brady Haran, Darts in Higher Dimensions, Numberphile video (2019)
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A006784, A160510 (decimal expansion), A320428 (continued fraction).

Digits:= 250:
engel:= (r, n)-> `if`(n=0 or r=0, NULL,
        [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
engel(evalf(exp(Pi/4)), 32);

A059177 Engel expansion of sqrt(10) = 3.16227...

Original entry on oeis.org

1, 1, 1, 7, 8, 12, 20, 86, 94, 118, 160, 179, 287, 315, 22588, 49419, 66011, 80779, 651661, 1078390, 1093865, 4254100, 27153191, 108815387, 220864645, 798937058, 992296124, 2196903274, 17524412379, 22828187385

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. A010467.

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[10], 7!], 10] (* modified by G. C. Greubel, Dec 26 2016 *)

A059178 Engel expansion of 2^(1/3) = 1.25992.

Original entry on oeis.org

1, 4, 26, 32, 58, 1361, 4767, 22303, 134563, 188609, 282816, 979804, 1272032, 1330628, 3719474, 5039143, 12531368, 435451235, 5391276884, 6140156718, 24140682996, 30267765913, 56443830660, 176797839116, 645251112512

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. A002580.

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[2^(1/3), 7!], 10] (* modified by G. C. Greubel, Dec 26 2016 *)

A059179 Engel expansion of 3^(1/3) = 1.44225.

Original entry on oeis.org

1, 3, 4, 4, 5, 8, 9, 14, 63, 91, 132, 605, 753, 993, 17297, 26120, 28227, 43466, 123132, 3551445, 7243732, 13958201, 41249856, 194184556, 3261328035, 13339681270, 18470226192, 23831447862, 25356135862

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

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[3^(1/3), 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)

cp's OEIS Frontend

A067239 a(0)=1, a(n) = 8*n*(2*n-1).

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A091831 Pierce expansion of 1/sqrt(2).

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A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).

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A199880 Engel expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.

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Maple

A222482 Engel expansion of cos(1)/(1+cos(1)).

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Maple

Formula

A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

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A329912 Engel expansion of exp(Pi/4).

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Maple

A059177 Engel expansion of sqrt(10) = 3.16227...

A067239 a(0)=1, a(n) = 8n(2*n-1).