A074938 -id:A074938 - cp's OEIS frontend

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

1, 7, 19, 25, 55, 61, 73, 79, 163, 169, 181, 187, 217, 223, 235, 241, 487, 493, 505, 511, 541, 547, 559, 565, 649, 655, 667, 673, 703, 709, 721, 727, 1459, 1465, 1477, 1483, 1513, 1519, 1531, 1537, 1621, 1627, 1639, 1645, 1675, 1681, 1693, 1699

Offset: 1

Wolfdieter Lang, May 23 2000

Related to A005836. Gives boundaries of open intervals that have to be erased in the Cantor middle-third set construction.
Let g(n) = Sum_{i=0..n} (i*binomial(n+i,i)^3*binomial(n,i)^2) = A112035(n). Let b = {m>0 : g(m) != 0 (mod 3)}. Then b(n) = a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
Conjecture: Similarly to A191107, this increasing sequence is generated by the rules: a(1) = 1, and if x is in the sequence, then 3*x-2 and 3*x+4 are also in the sequence. - L. Edson Jeffery, Nov 17 2015

			k=1: (1/3, 2/3);
k=2: (1/9, 2/9), (7/9, 8/9);
k=3: (1/27, 2/27), (7/27, 8/27), (19/27, 20/27), (25/27, 26/27); ...

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for 3-automatic sequences.

Cf. A003278, A005836, A005823, A055247, A112035.

Cf. A191107.

(* (Conjectured) Choose rows large enough to guarantee that all terms < max are generated. *)
rows = 1000; max = 10^4; a[1] = {1}; i = 1; Do[a[n_] = {}; Do[If[1 < 3*a[n - 1][[k]] - 2 < max, AppendTo[a[n], 3*a[n - 1][[k]] - 2], Break]; If[3*a[n - 1][[k]] + 4 < max, AppendTo[a[n], 3*a[n - 1][[k]] + 4], Break], {k, Length[a[n - 1]]}]; If[a[n] == {}, Break, i++], {n, 2, 1000}]; a055246 = Take[Flatten[Table[a[n], {n, i}]], 48] (* L. Edson Jeffery, Nov 17 2015 *)
Join[{1}, 1 + 6 Accumulate[Table[(3^IntegerExponent[n, 2] + 1)/2, {n, 60}]]] (* Vincenzo Librandi, Nov 26 2015 *)

g(n)=sum(i=0,n,i*binomial(n+i,i)^3*binomial(n,i)^2);
for (i=1,2000,if(Mod(g(i),3)<>0,print1(i,",")))

a(n) = fromdigits(binary(n-1),3)*6 + 1; \\ Kevin Ryde, Apr 23 2021

def A055246(n): return int(bin(n-1)[2:],3)*6|1 # Chai Wah Wu, Jun 26 2025

a(n) = 1+6*A005836(n), n >= 1.
a(n) = 1+3*A005823(n), n >= 1.
a(n+1) = A074938(n) + A074939(n); A074938: odd numbers in A005836, A074939: even numbers in A005836. - Philippe Deléham, Jul 10 2005
Conjecture: a(n) = 2*A191107(n) - 1 = 6*A003278(n) - 5 = (a((2*n-1)*2^(k-1))+2)/3^k, k>0. - L. Edson Jeffery, Nov 25 2015

Edited by N. J. A. Sloane, Nov 20 2015: used first comment to give more precise definition, and edited a comment at the suggestion of L. Edson Jeffery.

A006996 a(n) = C(2n,n) mod 3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, James Propp

Removing 0's from the sequence gives Thue-Morse sequence A001285 : 1,2,0,2,1,0,0,0,0,2,1,0,1,2,..->1,2,2,1,2,1,1,2,... - Benoit Cloitre, Jan 04 2004
a(n) = 0 if n in A074940, a(n) = 1 if n in A074939, a(n) = 2 if n in A074938.
Central terms of the triangle in A083093. - Reinhard Zumkeller, Jul 11 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..2187=3^7
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences that are fixed points of mappings

Cf. A001285.
Cf. A074938, A074939, A074940.
Cf. A083093.
Cf. A000984, A005704.

a006996 n = a083093 (2 * n) n  -- Reinhard Zumkeller, Jul 11 2013

Table[ Mod[ Binomial[2n, n], 3], {n, 0, 104}] (* Or *)
Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 0, 0}, 1 -> {1, 2, 0}, 2 -> {2, 1, 0}})]}], {1}, 7] (* Robert G. Wilson v, Mar 28 2005 *)

a(n)=if(n==0, return(1)); if(vecmax(Set(digits(n,3)))>1, 0, 1 + n%2) \\ Charles R Greathouse IV, May 09 2016

from gmpy2 import digits
def A006996(n): return 0 if '2' in digits(n,3) else 1+(n&1) # Chai Wah Wu, Jun 26 2025

a(n) = A000984(n) mod 3.
a(n) = A005704(n) mod 3. - Benoit Cloitre, Jan 04 2004
A fixed point of the morphism : 1 -> 120, 2 -> 210, 0 -> 000. - Philippe Deléham, Jan 08 2004

A074939 Even numbers such that base 3 representation contains no 2.

Original entry on oeis.org

0, 4, 10, 12, 28, 30, 36, 40, 82, 84, 90, 94, 108, 112, 118, 120, 244, 246, 252, 256, 270, 274, 280, 282, 324, 328, 334, 336, 352, 354, 360, 364, 730, 732, 738, 742, 756, 760, 766, 768, 810, 814, 820, 822, 838, 840, 846, 850, 972, 976, 982, 984, 1000, 1002

Offset: 0

Benoit Cloitre, Oct 04 2002; Nov 15 2003

Even numbers in A005836; n such that binomial(2n,n) == 1 (mod 3).

Sum of an even number of distinct powers of 3. - Emeric Deutsch, Dec 03 2003

Amiram Eldar, Table of n, a(n) for n = 0..10000
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Intersection of A005843 and A005836.

Cf. A006996, A010060, A055246, A074938, A083095.

Select[2*Range[0,600],DigitCount[#,3,2]==0&] (* Harvey P. Dale, Dec 10 2016 *)

def A074939(n): return int(bin((n.bit_count()&1)+(n<<1))[2:],3) # Chai Wah Wu, Jun 26 2025

a(n) = A083094(n)/2; a(n) mod 3 = A010060(n); n such that coefficient of x^n equals 1 in Product_{k>=0} (1 - x^(3^k)).

a(n) + A074938(n) = A055246(n+1). - Philippe Deléham, Jul 10 2005

A246510 G.f.: Sum_{n>=0} x^n / (1-4x)^(2n+1) * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].

Original entry on oeis.org

1, 5, 36, 305, 2821, 27690, 282699, 2967285, 31785786, 345815975, 3808549531, 42360017130, 474990254821, 5362633500755, 60897115958286, 695012481567465, 7966829676299139, 91674042449673960, 1058486539560201051, 12258669983923625475, 142359286920427682046, 1657287004720545992505

Offset: 0

Paul D. Hanna, Aug 27 2014

nonn

a(n) == 1 (mod 3) iff n = A074939(k) for k>=0, where A074939 gives even numbers such that base 3 representation contains no 2.

a(n) == 2 (mod 3) iff n = A074938(k) for k>=0, where A074938 gives odd numbers such that base 3 representation contains no 2.

			G.f.: A(x) = 1 + 5*x + 36*x^2 + 305*x^3 + 2821*x^4 + 27690*x^5 +...

Cf. A246509, A246056, A074938, A074939.

Table[Sum[3^k * Sum[Binomial[n-k, k+j]^2 * Binomial[k+j, j]^2 * 4^j,{j,0,n-2*k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Oct 04 2014 *)

/* By definition: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-4*x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 3^k * x^k) * sum(k=0, m, binomial(m, k)^2 * 4^k * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* By a binomial identity: */
{a(n)=local(A=1); A=sum(m=0, n, x^m/(1-x)^(2*m+1) * sum(k=0, m, binomial(m, k)^2 * 4^(m-k) * 3^k * x^k) * sum(k=0, m, binomial(m, k)^2 * x^k) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * sum(j=0, k, binomial(k, j)^2 * 4^(k-j) * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

/* By a binomial identity: */
{a(n)=polcoeff(sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^2 * 4^(m-k) * sum(j=0, k, binomial(k, j)^2 * 3^j * x^j)+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))

/* Formula for a(n): */
{a(n)=sum(k=0, n\2, sum(j=0, n-2*k, 3^k * binomial(n-k, k+j)^2 * binomial(k+j, j)^2 * 4^j))}
for(n=0, 25, print1(a(n), ", "))

G.f.: Sum_{n>=0} x^n / (1-x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * 3^k * x^k].
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * 4^(k-j) * 3^j * x^j.
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k)^2 * 4^(n-k) * Sum_{j=0..k} C(k,j)^2 * 3^j * x^j.
a(n) = Sum_{k=0..[n/2]} 3^k * Sum_{j=0..n-2*k} C(n-k, k+j)^2 * C(k+j, j)^2 * 4^j.
a(n) ~ sqrt(36 + 29*sqrt(3) + 3*sqrt(423 + 232*sqrt(3))) *  (9/2 + sqrt(3) + 3/2*sqrt(9 + 4*sqrt(3)))^n / (8*Pi*n). - Vaclav Kotesovec, Oct 04 2014

cp's OEIS Frontend

A055246 At step number k >= 1 the 2^(k-1) open intervals that are erased from [0,1] in the Cantor middle-third set construction are I(k,n) = (a(n)/3^k, (1+a(n))/3^k), n=1..2^(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A006996 a(n) = C(2n,n) mod 3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

A074939 Even numbers such that base 3 representation contains no 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A246510 G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

Formula

A246510 G.f.: Sum_{n>=0} x^n / (1-4x)^(2n+1) * [Sum_{k=0..n} C(n,k)^2 * 3^k * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].