A081609 -id:A081609 - cp's OEIS frontend

A081606 Numbers having at least one 1 in their ternary representation.

1, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Mar 23 2003

Complement of A005823.
Integers m such that central Delannoy number A001850(m) == 0 (mod 3). - Emeric Deutsch and Bruce E. Sagan, Dec 04 2003
Integers m such that A026375(m) == 0 (mod 3). - Fabio Visonà, Aug 03 2023

Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p.25.
Mathematics Stack Exchange, Proof that integers m in this sequence are such that A026375(m) == 0 (mod 3).

Cf. A007089, A062756, A081609, A081605, A074940.

Select[Range[100],DigitCount[#,3,1]>0&] (* Harvey P. Dale, Nov 26 2022 *)

from itertools import count, islice
def A081606_gen(): # generator of terms
    a = 0
    for n in count(1):
        b = int(bin(n)[2:],3)<<1
        yield from range(a+1,b)
        a = b
A081606_list = list(islice(A081606_gen(),30)) # Chai Wah Wu, Oct 13 2023

from gmpy2 import digits
def A081606(n):
    def f(x):
        s = digits(x>>1,3)
        for i in range(l:=len(s)):
            if s[i]>'1':
                break
        else:
            return n+int(s,2)
        return n-1+(int(s[:i] or '0',2)+1<Chai Wah Wu, Oct 29 2024

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

A061392 a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Henry Bottomley, Apr 30 2001

Number of nonnegative integers < n having no 1 in their ternary representation. - Reinhard Zumkeller, Mar 23 2003; corrected by Henry Bottomley, Mar 24 2003

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
Wikipedia, Cantor function.

k appears A061393(k) times.
Cf. A007089, A062756, A081608, A081609, A081611.
Essentially the partial sums of A088917.

A061392 := proc(n) option remember; local a : if n <=1 then n else A061392(floor(n/3)) + A061392(ceil(n/3)) fi: end: seq(A061392(n), n=0..87); # Johannes W. Meijer, Jun 05 2011

a(n+1) + A081609(n) = n+1. - Reinhard Zumkeller, Mar 23 2003; corrected by Henry Bottomley, Mar 24 2003
From Johannes W. Meijer, Jun 05 2011: (Start)
a(3*n+1) = a(n+1) + a(n), a(3*n+2) = a(n+1) + a(n) and a(3*n+3) = 2*a(n+1), for n>=1, with a(0)=0, a(1)=1, a(2)=1 and a(3)=2. [Northshield]
G.f.: x*Product_{n>=0} (1 + x^(3^n) + 2*x^(2*3^n) + x^(3*3^n) + x^(4*3^n)). [Northshield] (End)
Apparently, for any n >= 0 and k such that n < 3^k, a(n) = 2^k * c(n / 3^k) where c is the Cantor function. - Rémy Sigrist, Jul 12 2019

A081610 Number of numbers <= n having at least one 2 in their ternary representation.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 19, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 25, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset: 0

Reinhard Zumkeller, Mar 23 2003

a(n) + A081611(n) = n+1. Partial sums of A189820.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A007089, A081603, A081607, A081609, A189820.

num2tern := proc(n) return numboccur(convert(n,base,3),2): end: a:=0: for n from 0 to 80 do a:=a+`if`(num2tern(n)>0,1,0): printf("%d, ",a): od: # Nathaniel Johnston, May 17 2011

Accumulate[Table[If[DigitCount[n,3,2]>0,1,0],{n,0,70}]] (* Harvey P. Dale, Aug 20 2012 *)

first(n)=my(s,t); vector(n,k,t=Set(digits(k,3)); s+=t[#t]==2) \\ Charles R Greathouse IV, Sep 02 2015

from gmpy2 import digits
def A081610(n):
    l = (s:=digits(n,3)).find('2')
    if l >= 0: s = s[:l]+'1'*(len(s)-l)
    return n-int(s,2) # Chai Wah Wu, Dec 05 2024

a(n) ~ n. - Charles R Greathouse IV, Sep 02 2015

A081607 Number of numbers <= n having at least one 0 in their ternary representation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 26, 26, 27, 27, 27, 28, 29, 30, 31, 31, 31, 32, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 45, 46, 46, 46, 47, 48, 49, 50

Offset: 0

Reinhard Zumkeller, Mar 23 2003

nonn

a(n) + A081608(n) = n+1.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A007089, A077267, A081609, A081610.

f:= n -> `if`(has(convert(n,base,3),0),1,0):
ListTools:-PartialSums(map(f, [$0..100])); # Robert Israel, Mar 18 2018

Accumulate[Boole[Table[DigitCount[n,3,0]>0,{n,0,80}]]] (* Harvey P. Dale, Jun 23 2017 *)

first(n)=my(s,t); vector(n,k, t=Set(digits(k,3)); s+=(t[1]==0)) \\ Charles R Greathouse IV, Sep 02 2015

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A081606 Numbers having at least one 1 in their ternary representation.

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A061392 a(n) = a(floor(n/3)) + a(ceiling(n/3)) with a(0) = 0 and a(1) = 1.

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A081610 Number of numbers <= n having at least one 2 in their ternary representation.

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A081607 Number of numbers <= n having at least one 0 in their ternary representation.

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Maple

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