A007404 -id:A007404 - cp's OEIS frontend

A076214 Decimal expansion of C = Sum_{k>=0} 1/2^(2^k-1).

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

Offset: 1

Benoit Cloitre, Nov 03 2002

This constant has a nice continued fraction expansion (i.e. only 1 and 2 occur). C arises when looking for a sequence b(n) such that : b(1) = 0, b(n+1) is the smallest integer > b(n) such that the continued fraction for 1/2^b(1) + 1/2^b(2) + ... + 1/2^b(n+1) contains only 1's or 2's. Because b(n) = 2^n-1 and C = Sum_{k>=0} 1/2^b(k).

			1.632843018043786287416159475061050443406622751841105608682421807686111...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Boris Adamczewski, The Many Faces of the Kempner Number, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.15.
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.

Cf. A006466 (continued fraction), A007404, A078585.

Take[ RealDigits[ 2*NSum[1/2^2^k, {k, 0, Infinity}, WorkingPrecision -> 120]][[1]], 105] (* Jean-François Alcover, Nov 15 2011 *)

default(realprecision, 20080); x=suminf(k=0, 1/2^(2^k)); x*=2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b076214.txt", n, " ", d)); \\ Harry J. Smith, May 09 2009

Equals 2 * Sum_{k>=0} 1/2^(2^k) = 2 * A007404. - Harry J. Smith, May 09 2009
From Amiram Eldar, Mar 12 2024: (Start)
Equals 1 + 2 * A078585.
Equals 1 + Sum_{k>=1} floor(log_2(k))/2^k (Shamos, 2011, p. 8). (End)

A124091 Decimal expansion of Fibonacci binary constant: Sum{i>=0} (1/2)^Fibonacci(i).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 25 2006

This constant is transcendental, see A084119. - Charles R Greathouse IV, Nov 12 2014

			2.4102787972078658917940430244710631444834239245952787725932...

Harry J. Smith, Table of n, a(n) for n = 1..20000
D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Index entries for transcendental numbers

Cf. A007404 (Kempner-Mahler number), A125600 (continued fraction), A084119 (essentially the same).

Cf. A000301.

RealDigits[ N[ Sum[(1/2)^Fibonacci[i], {i, 0, Infinity}], 111]][[1]] (* Robert G. Wilson v, Nov 26 2006 *)

a=0 ; for(n=0,30, a += .5^fibonacci(n) ; print(a) ; )

default(realprecision, 20080); x=suminf(k=0, 1/2^fibonacci(k)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b124091.txt", n, " ", d)) \\ Harry J. Smith, May 04 2009

Equals Sum_{i>=0} 1/2^A000045(i).

Equals A084119 + 1.

More terms from Robert G. Wilson v, Nov 26 2006

A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)C(n,s)(C(n-s,s+1) - C(n-s-2,s-1)).

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 9, 5, 1, 14, 21, 1, 20, 56, 14, 1, 27, 120, 84, 1, 35, 225, 300, 42, 1, 44, 385, 825, 330, 1, 54, 616, 1925, 1485, 132, 1, 65, 936, 4004, 5005, 1287, 1, 77, 1365, 7644, 14014, 7007, 429

Offset: 3

Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007

Whereas A005043 counts certain trees, or noncrossed partitions, this subdivides the counts according to the number of leaves, or the lattice rank. Analogous to the Narayana triangle (A001263), where rows sum to the Catalan numbers.
Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012
Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. See also A108263 and A100754. - Tom Copeland, Oct 19 2014

			A005043(6) = 15 = 1+9+5 since NC (noncrossed, planar) partitions of 6-point cycle without singletons have 1,9,5 items with 1,2,3 blocks.
Triangle begins:
  1;
  1,   2;
  1,   5;
  1,   9,   5;
  1,  14,  21;
  1,  20,  56,  14;
  1,  27, 120,  84;
  1,  35, 225, 300,  42;
  1,  44, 385, 825, 330;
  ...

C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.

Row sums are A007404.

Cf. A001263, A005043, A033282, A100754, A108263, A108767, A132081.

/* triangle excluding 0 */ [[Binomial(n,k)*Binomial(n-2-k,k)/(k+1): k in [0..n-3]]: n in [3..15]]; // Vincenzo Librandi, Oct 19 2014

Map[Most, Table[(1/n) Binomial[n, s] (Binomial[n - s, s + 1] - Binomial[n - s - 2, s - 1]), {n, 3, 14}, {s, 0, n}] /. k_ /; k <= 0 -> Nothing] // Flatten (* Michael De Vlieger, Jan 09 2016 *)

a(n,k) = binomial(n,k)*binomial(n-2-k,k)/(k+1). - David Callan, Jul 22 2008
From Peter Bala, Oct 22 2008: (Start)
O.g.f.: (1 + x - sqrt(1 - 2*x + x^2*(1 - 4*a)))/(2*x*(1 + a*x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . [corrected by Jason Yuen, Sep 22 2024]
Define a functional I on formal power series of the form f(x) = 1 + a*x + b*x^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
Let now f(x) = 1 + a*x^2 + a*x^3 + a*x^4 + ... . Then the o.g.f. for this table is I(f(x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . Cf. A001263 and A108767. (End)

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

Name corrected by Emeric Deutsch, Dec 20 2014

A073414 Numerator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

Original entry on oeis.org

0, 1, 4, 9, 40, 169, 1054, 4385, 9824, 43681, 271910, 587501, 2621914, 16318985, 67897854, 287910401, 643718656, 2862785025, 17820428806, 38503642637, 171834999354, 725843640053, 4526896839672, 18833430998741, 42193758837154

Offset: 1

Benoit Cloitre, Aug 23 2002

Robert Israel, Table of n, a(n) for n = 1..1650

Cf. A007400, A007404, A073415.

a007400:= proc(n) option remember; local n8, n16;
    n8:= n mod 8;
    if n8 = 0 or n8 = 3 then return 2
    elif n8 = 4 or n8 = 7 then return 4
    elif n8 = 1 then return procname((n+1)/2)
    elif n8 = 2 then return procname((n+2)/2)
    fi;
    n16:= n mod 16;
    if n16 = 5 or n16 = 14 then return 4
    elif n16 = 6 or n16 = 13 then return 6
    fi
end proc:
a007400(0):= 0: a007400(1):= 1: a007400(2):= 4:
A[1]:= 0: A[2]:= 1:
for n from 3 to 100 do
  A[n]:= A[n-1]*a007400(n-1)+A[n-2];
od:
seq(A[n],n=1..100); # Robert Israel, Jun 14 2016

(* b is a007400 *)
b[n_] := b[n] = Module[{n8, n16}, n8 = Mod[n, 8]; Which[n8 == 0 || n8 == 3, Return[2], n8 == 4 || n8 == 7, Return[4], n8 == 1, Return[b[(n+1)/2]], n8 == 2, Return[b[(n+2)/2]]]; n16 = Mod[n, 16]; Which[n16 == 5 || n16 == 14, Return[4], n16 == 6 || n16 == 13, Return[6]]];
b[0] = 0; b[1] = 1; b[2] = 4;
a[1] = 0; a[2] = 1;
a[n_] := a[n] = a[n-1] b[n-1] + a[n-2];
Array[a, 100] (* Jean-François Alcover, Jun 10 2020, after Robert Israel *)

a(n)=component(component(contfracpnqn(contfrac(sum(k=0,20,1/2^(2^k)),n)),1),1)

a(n) = a(n-1) A007400(n-1) + a(n-2). - Robert Israel, Jun 14 2016

A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

Original entry on oeis.org

1, 1, 5, 11, 49, 207, 1291, 5371, 12033, 53503, 333051, 719605, 3211471, 19988431, 83165195, 352649211, 788463617, 3506503679, 21827485691, 47161475061, 210473385935, 889055018801, 5544803498741, 23068269013765, 51681341526271

Offset: 1

Benoit Cloitre, Aug 23 2002

Robert Israel, Table of n, a(n) for n = 1..1650

Cf. A007400, A007404, A073414.

a007400:= proc(n) option remember; local n8, n16;
    n8:= n mod 8;
    if n8 = 0 or n8 = 3 then return 2
    elif n8 = 4 or n8 = 7 then return 4
    elif n8 = 1 then return procname((n+1)/2)
    elif n8 = 2 then return procname((n+2)/2)
    fi;
    n16:= n mod 16;
    if n16 = 5 or n16 = 14 then return 4
    elif n16 = 6 or n16 = 13 then return 6
    fi
end proc:
a007400(0):= 0: a007400(1):= 1: a007400(2):= 4:
A[1]:= 1: A[2]:= 1:
for n from 3 to 100 do
  A[n]:= A[n-1]*a007400(n-1)+A[n-2];
od:
seq(A[n], n=1..100); # Robert Israel, Jun 14 2016

(* b is a007400 *)
b[n_] := b[n] = Module[{n8, n16}, n8 = Mod[n, 8]; Which[n8 == 0 || n8 == 3, Return[2], n8 == 4 || n8 == 7, Return[4], n8 == 1, Return[b[(n+1)/2]], n8 == 2, Return[b[(n+2)/2]]]; n16 = Mod[n, 16]; Which[n16 == 5 || n16 == 14, Return[4], n16 == 6 || n16 == 13, Return[6]]];
b[0] = 0; b[1] = 1; b[2] = 4;
a[1] = a[2] = 1;
a[n_] := a[n] = a[n-1] b[n-1] + a[n-2];
Array[a, 100] (* Jean-François Alcover, Jun 10 2020, after Robert Israel *)

a(n)=component(component(contfracpnqn(contfrac(sum(k=0,20,1/2^(2^k)),n)),1),2)

a(n) = a(n-1) A007400(n) + a(n-2). - Robert Israel, Jun 14 2016

A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, May 11 2025

The real root of the cubic polynomial 729*x^3 - 810*x^2 - 429*x + 233 matches this constant to 20 decimal places.

			0.3704211756339267984957431894112681...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2004(2020), page 9, eq. (3.1h), put q = 1/3 and alpha = 3. - _Amiram Eldar_, May 11 2025

Cf. A007404, A055777, A383818, A383819, A383820.

sum(k=1,logint(2^getlocalbitprec(),3)+1,moebius(3*k)/(3.^k - 1),0.) \\ Bill Allombert

Equals Sum_{k>=0} 1/3^(3^k) = Sum_{k>=0} 1/A055777(k). - Amiram Eldar, May 11 2025

A081846 Maximal element in the continued fraction for 1/n*sum(k>=0,1/2^(2^k)).

Original entry on oeis.org

6, 12, 19, 25, 33, 39, 46, 52, 60, 66, 72, 79, 85, 93, 99, 106, 112, 120, 126, 132, 139, 145, 153, 159, 166, 172, 180, 186, 192, 199, 205, 206, 219, 226, 232, 240, 246, 252, 259, 265, 273, 279, 286, 292, 300, 306, 313, 319, 326, 333, 339, 346, 352, 360, 366, 373, 379, 386, 393, 399, 406, 412, 420, 413, 433, 439, 446, 453, 459, 466, 472, 480, 486, 493, 499, 506, 513, 519, 526, 532, 540, 546, 553, 559, 567, 573, 579, 586, 593, 600, 606, 613, 619, 627, 633, 619, 646, 653, 660, 666

Offset: 1

Benoit Cloitre, Apr 10 2003

nonn

It seems that a(n)=20n/3 for infinitely many values of n.

Cf. A007400, A007404, A078816 (erroneous version).

Cf. A384939.

s = N[Sum[1/2^(2^k), {k, 0, Infinity}], 1000000]; Table[Max[ContinuedFraction[s/n]], {n, 1, 100}] (* Vaclav Kotesovec, Jul 22 2025 *)

Corrected and extended by Vaclav Kotesovec, Jul 22 2025

A275975 Decimal expansion of Sum_{k>=0}((-1)^k/2^(2^k)).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Aug 15 2016

Except for the alternating signs, this constant is defined in a similar way to the Kempner-Mahler number A007404. It is related to the Jeffreys binary sequence A275973 somewhat like Kempner-Mahler number is related to the Fredholm-Rueppel sequence A036987.
Conjecture: Numbers of the type Sum_{k>=0}(x^(2^k)) with algebraic x and |x|<1 are known to be transcendental (Mahler 1930, Adamczewski 2013). It is likely that the alternating sign does not invalidate this property.
Yes, this number is transcendental.  It is among various such forms Kempner showed are transcendental. - Kevin Ryde, Jul 12 2019

			0.308609008556231856400340479718025221697433904166441366801367221...

Boris Adamczewski, The Many Faces of the Kempner Number, arXiv:1303.1685 [math.NT], 2013.
Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society 17 (1916), pp. 476-482.
Kurt Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen, Math. Z. 32 (1930), 545-585.
Index entries for transcendental numbers.

Cf. A030300 (binary expansion), A160386.

Cf. A007404, A036987, A275973.

RealDigits[N[Sum[((-1)^k/2^(2^k)), {k, 0, Infinity}], 120]][[1]] (* Amiram Eldar, Jun 11 2023 *)

default(realprecision,2100);suminf(k=0,(-1)^k*0.5^2^k)

A356022 Decimal expansion of the infinite series Sum_{j>=0} 1/2^^j where ^^ indicates tetration.

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Jul 23 2022

This constant converges very fast to a finite value around 118785/65536, since tetration (AKA hyper-4) makes 2^^k become very big so quickly, even for "relatively small" values of k (e.g., 1/2^^5 < 1/10^19728.30).

This sequence has been inspired by a question posted by the user Max Muller on MathOverflow on July 2022 (see Links).

			1.812515258789...

MathOverflow, References on infinite series involving the tetration operator, like Sum_{n=1..oo} 1/2^^n, July 23 2022.

Cf. A014221, A356023, A007404.

RealDigits[Total[1/NestList[2^# &, 1, 5]], 10, 100][[1]] (* Amiram Eldar, Jul 23 2022 *)

Equals Sum_{j>=0} 1/A014221(j).

A356241 a(n) is the number of distinct Fermat numbers dividing n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1

Offset: 1

Amiram Eldar, Jul 30 2022

nonn

A051179(n) is the least number k such that a(k) = n.
The asymptotic density of occurrences of 0 is 1/2.
The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/2^(2^k) = (1/2) * A007404 = 0.4082107545... .

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Number.
Wikipedia, Fermat number.

Cf. A000215, A007404, A051158, A051179, A356242.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Count[f, _?(Divisible[n, #] &)]; Array[a, 100]

a(A000215(n)) = 1.
a(A051179(n)) = n.
a(A003593(n)) = A112753(n).
a(n) <= A356242(n).
a(A080307(n)) > 0 and a(A080308(n)) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)+1) = 0.5960631721... (A051158).

cp's OEIS Frontend

A076214 Decimal expansion of C = Sum_{k>=0} 1/2^(2^k-1).

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A124091 Decimal expansion of Fibonacci binary constant: Sum{i>=0} (1/2)^Fibonacci(i).

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A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)*C(n,s)*(C(n-s,s+1) - C(n-s-2,s-1)).

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A073414 Numerator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

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A073415 Denominator of the n-th convergent to Sum_{k>=0} 1/2^(2^k).

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A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

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A081846 Maximal element in the continued fraction for 1/n*sum(k>=0,1/2^(2^k)).

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A132081 Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)C(n,s)(C(n-s,s+1) - C(n-s-2,s-1)).