A006996 -id:A006996 - cp's OEIS frontend

A074940 Numbers having at least one 2 in their ternary representation.

2, 5, 6, 7, 8, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 29, 32, 33, 34, 35, 38, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 86, 87, 88, 89, 92

Offset: 1

Benoit Cloitre and Reinhard Zumkeller, Oct 04 2002; revised Dec 03 2003

Also, numbers m such that 3 divides C(2m,m).
Also, numbers m such that the central trinomial coefficient A002426(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Also, numbers m such that A092255(m) == 0 (mod 3). - Benoit Cloitre, Mar 22 2004
Also, numbers m such that the coefficient of x^m equals 0 in Product_{k>=0} (1-x^(3^k)). - N. J. A. Sloane, Jun 01 2010

			12 is not in the sequence since it is 110_3, but 11 is in the sequence since it is 102_3. - _Michael B. Porter_, Jun 30 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004.
Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Complement of A005836.
Cf. A006996, A007089, A081603, A081610, A081605, A081606.
A039966(a(n)) = 0.

a074940 n = a074940_list !! (n-1)
a074940_list = filter ((== 0) . a039966) [0..]
-- Reinhard Zumkeller, Jun 06 2012, Sep 29 2011

Select[Range@ 120, MemberQ[IntegerDigits[#, 3], 2] &] (* or *)
Select[Range@ 120, Divisible[Binomial[2 #, #], 3] &] (* Michael De Vlieger, Jun 29 2016 *)
Select[Range[100],DigitCount[#,3,2]>0&] (* Harvey P. Dale, Aug 25 2019 *)

is(n)=while(n,if(n%3==2,return(1));n\=3);0 \\ Charles R Greathouse IV, Aug 21 2011

from gmpy2 import digits
def A074940(n):
    def f(x):
        s = digits(x,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n+int(s[:i]+'1'*(l-i),2)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 29 2024

a(n) = n + O(n^0.631). - Charles R Greathouse IV, Aug 21 2011

More terms from Emeric Deutsch and Bruce E. Sagan, Dec 04 2003

A083093 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 3.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Apr 22 2003

Start with [1], repeatedly apply the map 0 -> [000/000/000], 1 -> [111/120/100], 2 -> [222/210/200]. - Philippe Deléham, Apr 16 2009

{T(n,k)} is a fractal gasket with fractal (Hausdorff) dimension log(A000217(3))/log(3) = log(6)/log(3) = 1.63092... (see Reiter reference). Replacing values greater than 1 with 1 produces a binary gasket with the same dimension (see Bondarenko reference). - Richard L. Ollerton, Dec 14 2021

			.            Rows 0 .. 3^3:
.    0:                             1
.    1:                            1 1
.    2:                           1 2 1
.    3:                          1 0 0 1
.    4:                         1 1 0 1 1
.    5:                        1 2 1 1 2 1
.    6:                       1 0 0 2 0 0 1
.    7:                      1 1 0 2 2 0 1 1
.    8:                     1 2 1 2 1 2 1 2 1
.    9:                    1 0 0 0 0 0 0 0 0 1
.   10:                   1 1 0 0 0 0 0 0 0 1 1
.   11:                  1 2 1 0 0 0 0 0 0 1 2 1
.   12:                 1 0 0 1 0 0 0 0 0 1 0 0 1
.   13:                1 1 0 1 1 0 0 0 0 1 1 0 1 1
.   14:               1 2 1 1 2 1 0 0 0 1 2 1 1 2 1
.   15:              1 0 0 2 0 0 1 0 0 1 0 0 2 0 0 1
.   16:             1 1 0 2 2 0 1 1 0 1 1 0 2 2 0 1 1
.   17:            1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1
.   18:           1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1
.   19:          1 1 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 1 1
.   20:         1 2 1 0 0 0 0 0 0 2 1 2 0 0 0 0 0 0 1 2 1
.   21:        1 0 0 1 0 0 0 0 0 2 0 0 2 0 0 0 0 0 1 0 0 1
.   22:       1 1 0 1 1 0 0 0 0 2 2 0 2 2 0 0 0 0 1 1 0 1 1
.   23:      1 2 1 1 2 1 0 0 0 2 1 2 2 1 2 0 0 0 1 2 1 1 2 1
.   24:     1 0 0 2 0 0 1 0 0 2 0 0 1 0 0 2 0 0 1 0 0 2 0 0 1
.   25:    1 1 0 2 2 0 1 1 0 2 2 0 1 1 0 2 2 0 1 1 0 2 2 0 1 1
.   26:   1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
.   27:  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 1 .
- _Reinhard Zumkeller_, Jul 11 2013

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
J.-P. Allouche, F. von Haeseler, H.-O. Peitgen, and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Disc. Appl. Math. 66 (1996) 1-22.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see pp. 130-132.
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
Lin Jiu and Christophe Vignat, On Binomial Identities in Arbitrary Bases, arXiv:1602.04149 [math.CO], 2016.
Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.
Y. Moshe, The distribution of elements in automatic double sequences, Discr. Math., 297 (2005), 91-103.
A. M. Reiter, Determining the dimension of fractals generated by Pascal's triangle, Fibonacci Quarterly, 31(2), 1993, pp. 112-120.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A051638 (row sums), A090044, A047999, A034931, A034930, A008975, A034932, A062296, A006047.
Cf. A006996 (central terms), A173019, A206424, A227428.
Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), (this sequence) (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

a083093 n k = a083093_tabl !! n !! k
a083093_row n = a083093_tabl !! n
a083093_tabl = iterate
   (\ws -> zipWith (\u v -> mod (u + v) 3) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Jul 11 2013

/* As triangle: */ [[Binomial(n,k) mod 3: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 15 2016

A083093 := proc(n,k)
    modp(binomial(n,k),3) ;
end proc:
seq(seq(A083093(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Jul 26 2017

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 3] (* Robert G. Wilson v, Jan 19 2004 *)

from sympy import binomial
def T(n, k):
    return binomial(n, k) % 3
for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Jul 26 2017

from math import comb, isqrt
def A083093(n):
    def f(m,k):
        if m<3 and k<3: return comb(m,k)%3
        c,a = divmod(m,3)
        d,b = divmod(k,3)
        return f(c,d)*f(a,b)%3
    return f(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2)) # Chai Wah Wu, Apr 30 2025

T(i, j) = binomial(i, j) mod 3.
T(n+1,k) = (T(n,k) + T(n,k-1)) mod 3. - Reinhard Zumkeller, Jul 11 2013
T(n,k) = Product_{i>=0} binomial(n_i,k_i) mod 3, where n = Sum_{i>=0} n_i*3^i and k = Sum_{i>=0} k_i*3^i, 0<=n_i, k_i <=2 [Allouche et al.]. - R. J. Mathar, Jul 26 2017

A005704 Number of partitions of 3n into powers of 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 23, 28, 33, 40, 47, 54, 63, 72, 81, 93, 105, 117, 132, 147, 162, 180, 198, 216, 239, 262, 285, 313, 341, 369, 402, 435, 468, 508, 548, 588, 635, 682, 729, 783, 837, 891, 954, 1017, 1080, 1152, 1224, 1296, 1377, 1458, 1539, 1632

Offset: 0

N. J. A. Sloane

nonn

Infinite convolution product of [1,2,3,3,3,3,3,3,3,3] aerated A000244 - 1 times, i.e., [1,2,3,3,3,3,3,3,3,3] * [1,0,0,2,0,0,3,0,0,3] * [1,0,0,0,0,0,0,0,0,2] * ... [Mats Granvik, Gary W. Adamson, Aug 07 2009]

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

T. D. Noe, Table of n, a(n) for n = 0..10000
G. E. Andrews, Congruence properties of the m-ary partition function, J. Number Theory 3 (1971), 104-110.
G. E. Andrews and J. A. Sellers, Characterizing the number of p-ary partitions modulo a prime p, p. 2.
C. Banderier, H.-K. Hwang, V. Ravelomanana, and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, 2012. - From _N. J. A. Sloane_, Dec 23 2012
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, Australasian J. Combin., 30 (2004), 193-196.
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions
D. Krenn, D. Ralaivaosaona, and S. Wagner, The Number of Multi-Base Representations of an Integer, 2014.
D. Krenn, D. Ralaivaosaona, and S. Wagner, Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems, arXiv:1503.08594 [math.NT], 2015. Also in Applicable Analysis and Discrete Mathematics (2015) Vol. 9, Issue 2, 285-312.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
O. J. Rodseth, Some arithmetical properties of m-ary partitions, Proc. Camb. Phil. Soc. 68 (1970), 447-453.
O. J. Rodseth and J. A. Sellers, On m-ary partition function congruences: A fresh look at a past problem, J. Number Theory 87 (2001), 270-281.
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

Cf. A000041, A000123, A005705, A005706, A018819.

Cf. A006996.

Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1, #1}]], #2]]] &, {1}, Range[2, 55]] (* Birkas Gyorgy, Apr 18 2011 *)
a[n_] := a[n] = If[n <= 2, n + 1, a[n - 1] + a[Floor[n/3]]]; Array[a, 101, 0] (* T. D. Noe, Apr 18 2011 *)

from functools import lru_cache
@lru_cache(maxsize=None)
def A005704(n): return A005704(n-1)+A005704(n//3) if n else 1 # Chai Wah Wu, Sep 21 2022

a(n) = a(n-1)+a(floor(n/3)).
Coefficient of x^(3*n) in prod(k>=0, 1/(1-x^(3^k))). Also, coefficient of x^n in prod(k>=0, 1/(1-x^(3^k)))/(1-x). - Benoit Cloitre, Nov 28 2002
a(n) mod 3 = binomial(2n, n) mod 3. - Benoit Cloitre, Jan 04 2004
Let T(x) be the g.f., then T(x)=(1-x^3)/(1-x)^2*T(x^3). [Joerg Arndt, May 12 2010]

Formula and more terms from Henry Bottomley, Apr 30 2001

A039969 An example of a d-perfect sequence: a(n) = Catalan(n) mod 3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

This is A006996 with all its terms repeated three times, except the initial term only twice. A006996 is a fixed point of the morphism 0 -> 000, 1 -> 120, 2 -> 210. [The original comment edited by Antti Karttunen, Aug 14 2017]
Equals Catalan(n) mod 3. (Cf. A000108.) - Paul D. Hanna, Jun 20 2003 [confirmed by Christian G. Bower, Jun 12 2005]
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.
David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link.
Index entries for sequences that are fixed points of mappings

Cf. A000108, A001006, A010872, A039972, A067397.

Cf. A006996 (trisection).

[Catalan(n) mod 3: n in [1..80]]; // Vincenzo Librandi, Jul 14 2015

seq(binomial(2*n, n)/(n+1) mod 3, n = 0 .. 100); # Robert Israel, Sep 20 2015

Take[ Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2, 0}, 2 -> {2, 1, 0}, 0 -> {0, 0, 0}}] &, {1}, 4] /. {1 -> {1, 1, 1}, 2 -> {2, 2, 2}, 0 -> {0, 0, 0}}], {2, 106}] (* or *)
Table[ Mod[ Binomial[ 2n, n]/(n + 1), 3], {n, 0, 104}] (* Robert G. Wilson v, Sep 09 2005 *)
Mod[CatalanNumber[Range[0,110]],3] (* Harvey P. Dale, Oct 23 2017 *)

A039969(n) = ((binomial(2*n, n)/(n+1))%3); \\ Antti Karttunen, Aug 13 2017

a(n) = ((-1)^(n+1)*A001006(n-1)) mod 3, for n>0. - Christian G. Bower, Jun 12 2005
a(n) = a(n-1) if n == 0 or 1 (mod 3). a(n) = 0 if n == 5,6, or 7 (mod 9). - Robert Israel, Sep 20 2015
a(3n) = A006996(n). - Antti Karttunen, Aug 14 2017
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021

More terms from Christian G. Bower, Jun 12 2005

Offset corrected from 1 to 0 by Antti Karttunen, Aug 13 2017

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

A081601 Numbers m such that 3 does not divide Sum_{k=0..m} binomial(2k,k) = A006134(m).

Original entry on oeis.org

0, 3, 9, 12, 27, 30, 36, 39, 81, 84, 90, 93, 108, 111, 117, 120, 243, 246, 252, 255, 270, 273, 279, 282, 324, 327, 333, 336, 351, 354, 360, 363, 729, 732, 738, 741, 756, 759, 765, 768, 810, 813, 819, 822, 837, 840, 846, 849, 972, 975, 981, 984, 999, 1002, 1008, 1011

Offset: 1

Benoit Cloitre, Apr 22 2003

Apparently a(n)/3 mod 2 = A010060(n-1), the Thue-Morse sequence.

a(n+1) is the smallest number with exactly n+1 partitions into distinct powers of 2 or of 3: A131996(a(n+1)) = n+1 and A131996(m) < n+1 for m < a(n+1). - Reinhard Zumkeller, Aug 06 2007

			For n=0, A006134(0) = 1, hence 0 is a term.

Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A005836, A006134, A006996, A010060, A083096, A131996.

Equals A089118(n-2) + 1, n > 1.

Select[Range[0, 1020], Mod[Sum[Binomial[2 k, k], {k, 0, #}], 3] != 0 &] (* Michael De Vlieger, Nov 28 2015 *)

for(n=0, 1e3, if(sum(k=0, n, binomial(2*k, k)) % 3 > 0, print1(n,", "))) \\ Altug Alkan, Nov 26 2015

Apparently a(n) = 3*A005836(n).

G.f.: (x/(1 - x))*Sum_{k>=0} 3^(k+1)*x^(2^k)/(1 + x^(2^k)) (conjecture). - Ilya Gutkovskiy, Jul 23 2017

Zero prepended to the sequence and formulas modified accordingly by L. Edson Jeffery, Nov 25 2015

A074938 Odd numbers such that base 3 representation contains no 2.

Original entry on oeis.org

1, 3, 9, 13, 27, 31, 37, 39, 81, 85, 91, 93, 109, 111, 117, 121, 243, 247, 253, 255, 271, 273, 279, 283, 325, 327, 333, 337, 351, 355, 361, 363, 729, 733, 739, 741, 757, 759, 765, 769, 811, 813, 819, 823, 837, 841, 847, 849, 973, 975, 981, 985, 999, 1003, 1009

Offset: 0

Benoit Cloitre, Oct 04 2002; Nov 15 2003

Odd numbers in A005836.
Numbers m such that coefficient of x^m equals -1 in Product_{k>=0} 1-x^(3^k).
Numbers k such that binomial(2k, k) == 2 (mod 3).
Sum of an odd 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; J. Num. Theory 117 (2006), 191-215.

Intersection of A005408 and A005836.

Cf. A006996, A074939.

Select[Range[1,1111,2],Count[IntegerDigits[#,3],2]==0&] (* Harvey P. Dale, Dec 19 2010 *)

def A074938(n): return int(bin((n<<1)+(n.bit_count()&1^1))[2:],3) # Chai Wah Wu, Jun 26 2025

a(n) (mod 3) = A010059(n).

((a(n)-1)/2) (mod 3) = A010060(n) = (1/2)*{binomial(2*a(n)+1, a(n)) (mod 3)}.

A039972 An example of a d-perfect sequence: a(n) = A007317(n) mod 3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

Odd bisection seems to be A006996. See also A039969. - Antti Karttunen, Aug 15 2017

Antti Karttunen, Table of n, a(n) for n = 1..6561
D. Kohel, S. Ling and C. Xing, Explicit Sequence Expansions, in Sequences and their Applications, C. Ding, T. Helleseth, and H. Niederreiter, eds., Proceedings of SETA'98 (Singapore, 1998), 308-317, 1999. DOI: 10.1007/978-1-4471-0551-0_23

Cf. A006996, A007317, A010872, A039969.

write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A039972_as_a_vector(size)= { my(A=vector(size)); for(j=1, size, A[j]=1+sum(k=1, j-1, A[k]*A[j-k])); apply(n->(n%3),A); }; \\ After Michael Somos' May 23 2005 code for A007317 and Christian G. Bower's formula for A039972.
write_to_bfile(1,A039972_as_a_vector(6561),"b039972_upto6561.txt"); \\ Antti Karttunen, Aug 15 2017

a(n) = A007317(n) mod 3. - Christian G. Bower, Jun 12 2005

More terms from Christian G. Bower, Jun 12 2005

Formula added to the name by Antti Karttunen, Aug 15 2017

A088564 a(n)=sum(i=0,n,binomial(2*i,i) (mod 3)).

Original entry on oeis.org

1, 3, 3, 5, 6, 6, 6, 6, 6, 8, 9, 9, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 15, 15, 16, 18, 18, 18, 18, 18, 19, 21, 21, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 0

Benoit Cloitre, Nov 19 2003

nonn

Distinct values (i.e. 1,3,5,6,8,9,...) are given by the partial sums of the Thue-Morse sequence on alphabet (1,2) A026430. Sequence of least k such that a(k)>a(k-1) is given by A005836. For any k>=0, card{ n : a(3*A005836(k)) =a(n)}=1.

Only 79 of the first 1001 terms are odd numbers. -- From Harvey P. Dale, Aug 08 2012

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001285, A005836, A006996.

Table[Sum[Mod[Binomial[2*i,i],3],{i,0,n}],{n,0,80}] (* Harvey P. Dale, Aug 08 2012 *)

cp's OEIS Frontend

A074940 Numbers having at least one 2 in their ternary representation.

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A083093 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 3.

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A005704 Number of partitions of 3n into powers of 3.

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A039969 An example of a d-perfect sequence: a(n) = Catalan(n) mod 3.

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

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A081601 Numbers m such that 3 does not divide Sum_{k=0..m} binomial(2k,k) = A006134(m).

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