A014081 -id:A014081 - cp's OEIS frontend

A239904 a(n) = n - wt(n) + (number of times 11 appears in binary expansion of n).

0, 0, 1, 2, 3, 3, 5, 6, 7, 7, 8, 9, 11, 11, 13, 14, 15, 15, 16, 17, 18, 18, 20, 21, 23, 23, 24, 25, 27, 27, 29, 30, 31, 31, 32, 33, 34, 34, 36, 37, 38, 38, 39, 40, 42, 42, 44, 45, 47, 47, 48, 49, 50, 50, 52, 53, 55, 55, 56, 57, 59, 59, 61, 62, 63, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71, 72, 74, 74, 76

Offset: 0

N. J. A. Sloane, Apr 06 2014

This is G_{2, 1/4}(n) in Prodinger's notation.

Gheorghe Coserea, Table of n, a(n) for n = 0..10000
Helmut Prodinger, Generalizing the sum of digits function, SIAM J. Algebraic Discrete Methods 3 (1982), no. 1, 35--42. MR0644955 (83f:10009).

Cf. A000120, A014081, A239906, A239907.

A000120 := proc(n) add(i, i=convert(n, base, 2)) end:
# A014081:
cn := proc(v, k) local n, s, nn, i, j, som, kk;
som := 0;
kk := convert(cat(seq(1, j = 1 .. k)),string);
n := convert(v, binary);
s := convert(n, string);
nn := length(s);
for i to nn - k + 1 do
if substring(s, i .. i + k - 1) = kk then som := som + 1 fi od;
som; end;
[seq(n-A000120(n)+cn(n,2), n=0..100)];

cn[n_, k_] := Count[Partition[IntegerDigits[n, 2], k, 1], Table[1, {k}]]; Table[n - DigitCount[n, 2, 1] + cn[n, 2], {n, 0, 78}] (* Michael De Vlieger, Sep 18 2015 *)

a(n) = n - hammingweight(n) + hammingweight(bitand(n, n>>1));
vector(79, i, a(i-1))  \\ Gheorghe Coserea, Sep 24 2015

def A239904(n): return n-n.bit_count()+(n&(n>>1)).bit_count() # Chai Wah Wu, Feb 12 2023

a(n) = n - A000120(n) + A014081(n).

A257739 Numbers n for which A256999(n) > n; numbers that can be made larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

Original entry on oeis.org

5, 9, 10, 11, 13, 17, 18, 19, 20, 21, 22, 23, 25, 27, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111

Offset: 1

Antti Karttunen, May 18 2015

Note that A256999(a(n)) is always in A257250.

If we define a co-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is not a co-necklace. Numbers whose binary expansion, without the most significant digit, is not a necklace are A329367. - Gus Wiseman, Nov 14 2019

			For n = 5 with binary representation "101" if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction we get "110" = 6 > 5, so 5 can be made larger by such rotations and thus 5 is included in this sequence.
For n = 6 with binary representation "110" no such rotation will yield a larger number and thus 6 is NOT included in this sequence.
For n = 10 with binary representation "1010" if we rotate other bits than the most significant bit (that is, only the three rightmost digits "010") either one step to the left or two steps to the right we get "1100" = 12 > 10, thus 10 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A257250.
Cf. A080541, A080542, A256999.
Numbers whose binary expansion is a necklace are A275692.
Numbers whose binary expansion is a co-necklace are A065609.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose non-msb expansion is a co-necklace are A257250.
Numbers whose non-msb expansion is a necklace are A328668.
Numbers whose reversed non-msb expansion is a necklace are A328607.
Numbers whose non-msb expansion is not a necklace are A329367.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A121016, A164707, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[2,100],!reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 14 2019 *)

A051032 Summatory Rudin-Shapiro sequence for 2^(n-1).

Original entry on oeis.org

2, 3, 3, 5, 5, 9, 9, 17, 17, 33, 33, 65, 65, 129, 129, 257, 257, 513, 513, 1025, 1025, 2049, 2049, 4097, 4097, 8193, 8193, 16385, 16385, 32769, 32769, 65537, 65537, 131073, 131073, 262145, 262145, 524289, 524289, 1048577, 1048577, 2097153

Offset: 1

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A014081, A020985, A020986, A000051.

d2n[n_]:=Module[{x=2^n+1},{x,x}];Join[{2},Flatten[Array[d2n,30]]] (* Harvey P. Dale, May 26 2011 *)

Apart from leading term, just A000051 (2^n + 1) doubled up.
G.f.: -x*(-2 - x + 4*x^2)/((x - 1)*(2*x^2 - 1)). - R. J. Mathar, Jul 15 2016
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3). - Wesley Ivan Hurt, Aug 19 2022
E.g.f.: cosh(x) + cosh(sqrt(2)*x) + sinh(x) + sinh(sqrt(2)*x)/sqrt(2) - 2. - Stefano Spezia, Feb 05 2023

A056966 In binary: write what is described (putting a leading zero on numbers which have an odd number of binary digits).

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jul 20 2000

			a(54)=2 because 54 = 110110 base 2, which can be read as one 1 followed by zero 1's followed by one 0, i.e., 10 base 2 = 2 base 10.

Cf. A014081, A049190, A056967.

def A056966(n):
    s = bin(n)[2:]
    s = '0'*(len(s)&1)+s
    return int('0'+''.join(s[i+1]*int(s[i])for i in range(0,len(s),2)),2) # Chai Wah Wu, Feb 12 2023

A100046 Decimal expansion of -Pi/4 + (3*log(2))/2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Oct 31 2004

			0.2543226074...

Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A003881, A014081, A262023.

RealDigits[(3*Log[2])/2-Pi/4,10,120][[1]] (* Harvey P. Dale, May 28 2018 *)

Equals Sum_{k>=1} A014081(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A196096 Occurrences of '11' in base 3 expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 2, 3, 2, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Jonathan Vos Post, Sep 27 2011

Occurrences of '11' in A007089(n). This is to base 3 and A007089 as A014081 is to base 2 A007088.

First occurrence of k>0 = 4, 13, 40, 121, 364, ..., = A003462(k). - Robert G. Wilson v, Sep 27 2011

			a(4) = 1 because 4 in base 3 is "11" which has one instance of "11".
a(13) = 2 because the number 13 in base 3 is "111" which has two substrings of "11".

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A007089, A014081.

A196096 := proc(n)
        local a,dgs3 ;
        a := 0 ;
        dgs3 := convert(n,base,3) ;
        for i from 1 to nops(dgs3)-1 do
                if op(i,dgs3)=1 and op(i+1,dgs3)=1 then
                        a := a+1 ;
                end if;
        end do;
        a ;
end proc:
seq(A196096(n),n=0..80) ; # R. J. Mathar, Sep 28 2011

f[n_] := Count[ Partition[ IntegerDigits[ n, 3], 2, 1], {1, 1}]; Array[f, 100, 0] (* Robert G. Wilson v, Sep 27 2011 *)

A270438 a(n) is the number of entries == 1 mod 4 in row n of Pascal's triangle.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 8, 2, 4, 4, 4, 4, 8, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4, 4, 4, 8, 4, 8, 4, 8, 8, 8, 4, 8, 8, 16, 4, 8, 8

Offset: 0

Robert Israel, Jul 12 2016

All entries are powers of 2.

			Row 3 of Pascal's triangle is (1,3,3,1) and has two entries == 1 (mod 4), so a(3) = 2.

Robert Israel, Table of n, a(n) for n = 0..10000
Kenneth S. Davis and William A. Webb, Pascal's triangle modulo 4, Fib. Quart., 29 (1991), 79-83.
A. Granville, Zaphod Beeblebrox's Brain and the Fifty-ninth Row of Pascal's Triangle, The American Mathematical Monthly, 99(4) (1992), 318-331.

Cf. A034931, A163000, A000120, A007318, A014081.

f:= proc(n) local L,m;
  L:= convert(n,base,2);
  m:= convert(L,`+`);
  if has(L[1..-2]+L[2..-1],2) then 2^(m-1) else 2^m fi
end proc:
map(f, [$0..1000]);

Count[#, 1] & /@ Table[Mod[Binomial[n, k], 4], {n, 0, 120}, {k, 0, n}] (* Michael De Vlieger, Feb 26 2017 *)

a(n) = 2^(hammingweight(n) - min(hammingweight(bitand(n, n>>1)),1)) \\ Charles R Greathouse IV, Jul 13 2016

def A270438(n): return 1<>1)).bit_count() # Chai Wah Wu, Apr 24 2025

a(n) = 2^(A000120(n) - min(1, A014081(n))). [Davis & Webb]

A328870 Numbers whose lengths of runs of 1's in their reversed binary expansion are not weakly increasing.

Original entry on oeis.org

11, 19, 22, 23, 35, 38, 39, 43, 44, 45, 46, 47, 55, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 103, 107, 110, 111, 131, 134, 135, 139, 140, 141, 142, 143, 147, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 163, 166, 167

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

			The sequence of terms together with their reversed binary expansions begins:
  11: (1101)
  19: (11001)
  22: (01101)
  23: (11101)
  35: (110001)
  38: (011001)
  39: (111001)
  43: (110101)
  44: (001101)
  45: (101101)
  46: (011101)
  47: (111101)
  55: (111011)
  67: (1100001)
  70: (0110001)
  71: (1110001)
  75: (1101001)
  76: (0011001)
  77: (1011001)
  78: (0111001)

Complement of A328869.
The version for prime indices is A112769.
The binary expansion of n has A069010(n) runs of 1's.
Cf. A000120, A003714, A014081, A164707, A245563, A304678, A328592.

Select[Range[100],!LessEqual@@Length/@Split[Join@@Position[Reverse[IntegerDigits[#,2]],1],#2==#1+1&]&]

A014083 Occurrences of '1111' in binary expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Simon Plouffe

			a(63) = 3 as 63 = 111111 in binary and 1111 occurs three times (different occurrences may overlap). - _Antti Karttunen_, Jul 24 2017

Antti Karttunen, Table of n, a(n) for n = 0..65536
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

Cf. A000120, A014081, A014082.

Maple
```
See A014081.
```

Table[SequenceCount[IntegerDigits[n,2],{1,1,1,1},Overlaps->True],{n,0,100}] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Sep 25 2015 *)

u1111(n)=my(v=binary(n)); sum(k=1,#v-3, v[k]&&v[k+1]&&v[k+2]&&v[k+3])

a(n)=my(s,t); while(n, n>>=valuation(n,2); t=valuation(n+1,2); s+=max(t-3, 0); n>>=t); s \\ Charles R Greathouse IV, Jan 21 2016

a(2n) = a(n), a(2n+1) = a(n) + [n congruent to 7 mod 8]. - Ralf Stephan, Aug 21 2003
G.f.: 1/(1-x) * sum(k>=0, t^15(1-t)/(1-t^16), t=x^2^k). - Ralf Stephan, Sep 08 2003
a(n) <= log_2(n+1) - 3 for n >= 7. - Charles R Greathouse IV, Jan 21 2016

Term a(0)=0 prepended and more terms from Antti Karttunen, Jul 24 2017

A196272 Number of occurrences of '11' in base-4 expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 3

Offset: 0

Jonathan Vos Post, Sep 29 2011

This is to base 4 A007090 as A196096 is to base 3 and A007089 as A014081 is to base 2 A007088.

Records occur at places where n = '111...11' in base b, that is n = (b^(k+1)-1)/(b-1) for some k > 0, in particular for b=4 as listed in A002450(k+1). - R. J. Mathar, Sep 30 2011

			a(21) = 2 because 21 (base 10) = 111 (base 4), whose first two digits are 1's, and whose rightmost two digits are the second substring of "11".

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A007090, A014081, A196096.

A196272 := proc(n)
        local a, dgs ;
        a := 0 ;
        dgs := convert(n, base, 4) ;
        for i from 1 to nops(dgs)-1 do
                if op(i, dgs)=1 and op(i+1, dgs)=1 then
                        a := a+1 ;
                end if;
        end do;
        a ;
end proc:
seq(A196272(n), n=0..100) ; # R. J. Mathar, Sep 30 2011

Table[d = IntegerDigits[n, 4]; Count[Partition[d, 2, 1], {1, 1}], {n, 0, 100}] (* T. D. Noe, Sep 30 2011 *)

cp's OEIS Frontend

A239904 a(n) = n - wt(n) + (number of times 11 appears in binary expansion of n).

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A257739 Numbers n for which A256999(n) > n; numbers that can be made larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

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Mathematica

A051032 Summatory Rudin-Shapiro sequence for 2^(n-1).

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Crossrefs

Programs

Mathematica

Formula

A056966 In binary: write what is described (putting a leading zero on numbers which have an odd number of binary digits).

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Keywords

Examples

Crossrefs

Programs

Python

A100046 Decimal expansion of -Pi/4 + (3*log(2))/2.

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Mathematica

Formula

A196096 Occurrences of '11' in base 3 expansion of n.

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Maple

Mathematica

A270438 a(n) is the number of entries == 1 mod 4 in row n of Pascal's triangle.

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Maple

Mathematica

PARI

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A328870 Numbers whose lengths of runs of 1's in their reversed binary expansion are not weakly increasing.

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