A003726 -id:A003726 - cp's OEIS frontend

A004742 Numbers whose binary expansion does not contain 101.

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 19, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 48, 49, 50, 51, 56, 57, 60, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 76, 78, 79, 96, 97, 98, 99, 100, 102, 103, 112, 113, 114, 115, 120, 121, 124, 126, 127

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004743 (no 110), A003726 (no 111).

a004742 n = a004742_list !! (n-1)
a004742_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 5 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0, 130], !StringContainsQ[IntegerString[#, 2], "101"] &] (* Amiram Eldar, Feb 13 2022 *)

is(n)=n=binary(n);for(i=3,#n,if(n[i]&&n[i-2]&&!n[i-1],return(0))); 1 \\ Charles R Greathouse IV, Mar 26 2013

is(n)=while(n>4, if(bitand(n,7)==5, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

is(n)=!bitand(bitand(n,n>>2),bitneg(n>>1)) \\ Charles R Greathouse IV, Oct 28 2021

searchLE(S,x)=my(t=setsearch(S,x)); if(t,t,setsearch(S,x,1)-1); \\ finds last element <= x
expand(~v, lim)=my(b=exponent(v[#v]+1), B=1<lim, listpop(~v));
list(lim)=lim\=1; if(lim<5, return(if(lim<0,[],[0..lim]))); my(v=List([0..3])); for(b=3,exponent(lim+1), expand(~v, 2^b-1)); expand(~v, lim); Vec(v)

Sum_{n>=2} 1/a(n) = 6.198475910942069028389983717965787117743378665090593775808705963863146498248... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004743 Numbers whose binary expansion does not contain 110.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 47, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 79, 80, 81, 82, 83, 84, 85, 87, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A003726 (no 111).

a004743 n = a004743_list !! (n-1)
a004743_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 6 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0, 140], !StringContainsQ[IntegerString[#, 2], "110"] &] (* Amiram Eldar, Feb 13 2022 *)
Select[Range[0,150],SequenceCount[IntegerDigits[#,2],{1,1,0}]==0&] (* Harvey P. Dale, Mar 14 2025 *)

is(n)=n=binary(n);for(i=3,#n,if(!n[i]&&n[i-2]&&n[i-1],return(0))); 1 \\ Charles R Greathouse IV, Mar 26 2013

is(n)=while(n>5, if(bitand(n,7)==6, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 5.126608057149204485684180689064467269298250594297584060475240185531109866051... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004744 Numbers whose binary expansion does not contain 011.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 48, 49, 50, 52, 53, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 96, 97, 98, 100, 101, 104, 105

Offset: 1

N. J. A. Sloane

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).

a004744 n = a004744_list !! (n-1)
a004744_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 3 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,110],!MemberQ[Partition[IntegerDigits[#,2],3,1],{0,1,1}]&] (* Harvey P. Dale, Oct 15 2013 *)

is(n)=n=binary(n);for(i=3,#n,if(n[i]&&n[i-1]&&!n[i-2], return(0)));1 \\ Charles R Greathouse IV, Mar 26 2013

is(n)=while(n>10, if(bitand(n,7)==3, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 6.084750966700965350831194838591995529232464122788387705746226526437263331240... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A117546 Number of representations of n as a sum of distinct tribonacci numbers (A000073).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 2, 2, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

T. D. Noe and Jonathan Vos Post, Mar 28 2006

It can be shown that, like the Fibonacci numbers, the tribonacci numbers are complete; that is, a(n)>0 for all n. There is always a representation, free of three consecutive tribonacci numbers, which is analogous to the Zeckendorf representation of Fibonacci numbers. See A003726.

			a(14)=2 because 14 is both 13+1 and 7+4+2+1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Tribonacci Number.
Eric Weisstein's World of Mathematics, Zeckendorf Representation.

Cf. A000119 (number of representations of n as a sum of distinct Fibonacci numbers).

Cf. A240844.

a117546 = p $ drop 3 a000073_list where
   p _  0     = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Apr 13 2014

tr={1,2,4,7,13,24,44,81,149}; len=tr[[ -1]]; cnt=Table[0,{len}]; Do[v=IntegerDigits[k,2,Length[tr]]; s=Dot[tr,v]; If[s<=len, cnt[[s]]++ ], {k,2^(Length[tr])-1}]; cnt

a(0)=1 added and offset changed by Reinhard Zumkeller, Apr 13 2014

A004745 Numbers whose binary expansion does not contain 001.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 40, 42, 43, 44, 45, 46, 47, 48, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 80, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 104, 106, 107, 108, 109, 110

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).

a004745 n = a004745_list !! (n-1)
a004745_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 1 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0, 110], ! StringContainsQ[IntegerString[#, 2], "001"] &] (* Amiram Eldar, Feb 13 2022 *)
Select[Range[0,120],SequenceCount[IntegerDigits[#,2],{0,0,1}]==0&] (* Harvey P. Dale, Jul 05 2024 *)

is(n)=n=binary(n);for(i=4,#n,if(n[i]&&!n[i-1]&&!n[i-2], return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

is(n)=while(n>8, if(bitand(n,7)==1, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 5.808784664093998434778841785199192904637860758506854276321167162567685504669... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004746 Numbers whose binary expansion does not contain 010.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 38, 39, 44, 45, 46, 47, 48, 49, 51, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 67, 70, 71, 76, 77, 78, 79, 88, 89, 91, 92, 93, 94, 95, 96, 97, 99, 102

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).

a004746 n = a004746_list !! (n-1)
a004746_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 2 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,110],SequenceCount[IntegerDigits[#,2],{0,1,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Oct 19 2015 *)

is(n)=n=binary(n);for(i=4,#n,if(!n[i]&&n[i-1]&&!n[i-2], return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

is(n)=while(n>9, if(bitand(n,7)==2, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 7.338340181978485860731253930056466995425939377143636935044890325770833657631... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A004781 Binary expansion contains 3 adjacent 1's.

Original entry on oeis.org

7, 14, 15, 23, 28, 29, 30, 31, 39, 46, 47, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 78, 79, 87, 92, 93, 94, 95, 103, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 142, 143, 151, 156, 157, 158, 159, 167

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 2-automatic sequences.

Complement of A003726.

Cf. A004779, A136037.

a004781 n = a004781_list !! (n - 1)
a004781_list = filter f [0..] where
   f x | x < 7     = False
       | otherwise = (x `mod` 8) == 7 || f (x `div` 2)
-- Reinhard Zumkeller, Jun 03 2012

q:= n-> verify([1$3], Bits[Split](n), 'sublist'):
select(q, [$0..200])[];  # Alois P. Heinz, Oct 22 2021

Select[Range[200],MemberQ[Partition[IntegerDigits[#,2],3,1], {1,1,1}]&]  (* Harvey P. Dale, Mar 31 2011 *)
Select[Range[200], StringContainsQ[IntegerString[#, 2], "111"] &] (* Amiram Eldar, Oct 22 2021 *)
Select[Range[200],SequenceCount[IntegerDigits[#,2],{1,1,1}]>0&] (* Harvey P. Dale, Dec 28 2021 *)

is(n)=!!bitand(bitand(n,n<<1),n<<2) \\ Charles R Greathouse IV, Sep 24 2012

a(n) ~ n. - Charles R Greathouse IV, Sep 24 2012

Offset corrected by Reinhard Zumkeller, Jun 03 2012

A003265 Not representable by truncated tribonacci sequence 2, 4, 7, 13, 24, 44, 81, ....

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 32, 34, 36, 38, 40, 42, 45, 47, 49, 52, 54, 56, 58, 60, 62, 65, 67, 69, 71, 73, 76, 78, 80, 82, 84, 86, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 123

Offset: 1

N. J. A. Sloane

nonn

The usual tribonacci representation of n writes n as a sum of tribonacci numbers 1, 2, 4, 7, 13, 24, ... (A000073) avoiding using three consecutive numbers (see A003726, A278038). But if we are not allowed to use 1, then some numbers cannot be represented, and such numbers are listed here. - N. J. A. Sloane, Oct 08 2018
Indices of odd terms of A003726. - Charlie Neder, Apr 25 2019
Numbers whose tribonacci representation ends in 1. Equivalently, the first column of the Trithoff (tribonacci) array, see A136175. - Tanya Khovanova and PRIMES STEP Senior group, May 07 2022

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charlie Neder, Table of n, a(n) for n = 1..1000
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 65.

Cf. A000073, A003726, A136175, A278038.

Edited by N. J. A. Sloane, Oct 08 2018

a(47)-a(57) from Charlie Neder, Apr 25 2019

A356964 Replace 2^k in binary expansion of n with tribonacci(k+3) (where tribonacci corresponds to A000073).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Sep 06 2022

This sequence is to tribonacci numbers (A000073) what A022290 is to Fibonacci numbers (A000045).

For any k >= 0, k appears A117546(k) times in this sequence.

			For n = 9:
- 9 = 2^3 + 2^0,
- so a(9) = A000073(3+3) + A000073(0+3) = 7 + 1 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000073, A003726, A003796, A022290, A117546.

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n,2); v+=([0,1,0; 0,0,1; 1,1,1]^(3+k))[2,1]); return (v); }

def A356964(n):
    a, b, c, s = 1,2,4,0
    for i in bin(n)[-1:1:-1]:
        s += int(i)*a
        a, b, c = b, c, a+b+c
    return s # Chai Wah Wu, Sep 10 2022

a(A003726(n+1)) = n.

a(A003796(n+1)) = n.

A305377 Tribonacci representation of primes, written in base 10.

Original entry on oeis.org

2, 3, 5, 8, 12, 16, 20, 22, 27, 37, 40, 48, 52, 54, 67, 74, 82, 84, 91, 99, 101, 108, 130, 137, 147, 152, 154, 162, 164, 169, 194, 198, 205, 209, 256, 258, 265, 273, 277, 288, 294, 297, 309, 320, 324, 326, 341, 358, 363, 365, 387, 394, 396, 409, 419, 426, 434, 436, 515, 520, 522, 534, 554, 560

Offset: 1

N. J. A. Sloane, Jun 12 2018

Robert Israel, Table of n, a(n) for n = 1..10001

Equals A003726(prime(n)).

Cf. A278038, A305379.

L[0]:= [0]: L[1]:= [1]:
for d from 2 to 15 do
  L[d]:= map(t -> (2*t, `if`(t mod 4 <> 3, 2*t+1,NULL)), L[d-1])
od:
A003726:=map(op,[seq(L[i],i=0..15)]):
seq(A003726[ithprime(i)+1],i=1..numtheory:-pi(nops(A003726)-1)); # Robert Israel, Jun 12 2018

from sympy import prime
def A305377(n):
    m, tlist, s = prime(n), [1,2,4], 0
    while tlist[-1]+tlist[-2]+tlist[-3] <= m:
        tlist.append(tlist[-1]+tlist[-2]+tlist[-3])
    for d in tlist[::-1]:
        s *= 2
        if d <= m:
            s += 1
            m -= d
    return s # Chai Wah Wu, Jun 12 2018

More terms from Robert Israel, Jun 12 2018

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A004742 Numbers whose binary expansion does not contain 101.

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A004743 Numbers whose binary expansion does not contain 110.

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A004744 Numbers whose binary expansion does not contain 011.

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A117546 Number of representations of n as a sum of distinct tribonacci numbers (A000073).

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A004745 Numbers whose binary expansion does not contain 001.

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A004746 Numbers whose binary expansion does not contain 010.

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A004781 Binary expansion contains 3 adjacent 1's.

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