A084471 -id:A084471 - cp's OEIS frontend

A048678 Binary expansion of nonnegative integers expanded to "Zeckendorffian format" with rewrite rules 0->0, 1->01.

0, 1, 2, 5, 4, 9, 10, 21, 8, 17, 18, 37, 20, 41, 42, 85, 16, 33, 34, 69, 36, 73, 74, 149, 40, 81, 82, 165, 84, 169, 170, 341, 32, 65, 66, 133, 68, 137, 138, 277, 72, 145, 146, 293, 148, 297, 298, 597, 80, 161, 162, 325, 164, 329, 330, 661, 168, 337, 338, 677, 340

Offset: 0

Antti Karttunen

No two adjacent 1-bits. Permutation of A003714.

Replace 1 with 01 in binary. - Ralf Stephan, Oct 07 2003

			11=1011 in binary, thus is rewritten as 100101 = 37 in decimal.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A003714, A005203, A048679, A048680.
MASKTRANS transform of A053644.
Cf. A084471, A088697, A088698.
Cf. A124108.

a048678 0 = 0
a048678 x = 2 * (b + 1) * a048678 x' + b
            where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 31 2015

rewrite_0to0_1to01 := proc(n) option remember; if(n < 2) then RETURN(n); else RETURN(((2^(1+(n mod 2))) * rewrite_0to0_1to01(floor(n/2))) + (n mod 2)); fi; end;

f[n_] := FromDigits[ Flatten[IntegerDigits[n, 2] /. {1 -> {0, 1}}], 2]; Table[f@n, {n, 0, 60}] (* Robert G. Wilson v, Dec 11 2009 *)

a(n)=if(n<1,0,(3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2)

a(n) = if(n == 0, 0, my(A = -2); sum(i = 0, logint(n, 2), A++; if(bittest(n, i), 1 << (A++)))) \\ Mikhail Kurkov, Mar 14 2024

def a(n):
    return 0 if n==0 else (3 - (-1)**n)*a(n//2) + (1 - (-1)**n)//2
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017

def A048678(n): return int(bin(n)[2:].replace('1','01'),2) # Chai Wah Wu, Mar 18 2024

a(n) = rewrite_0to0_1to01(n) [ Each 0->1, 1->10 in binary expansion of n ].
a(0)=0; a(n) = (3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2. - Benoit Cloitre, Aug 31 2003
a(0)=0, a(2n) = 2a(n), a(2n+1) = 4a(n) + 1. - Ralf Stephan, Oct 07 2003

A175047 Write n in binary, then increase each run of 0's by one 0. a(n) is the decimal equivalent of the result.

Original entry on oeis.org

1, 4, 3, 8, 9, 12, 7, 16, 17, 36, 19, 24, 25, 28, 15, 32, 33, 68, 35, 72, 73, 76, 39, 48, 49, 100, 51, 56, 57, 60, 31, 64, 65, 132, 67, 136, 137, 140, 71, 144, 145, 292, 147, 152, 153, 156, 79, 96, 97, 196, 99, 200, 201, 204, 103, 112, 113, 228, 115, 120, 121, 124, 63, 128

Offset: 1

Leroy Quet, Dec 02 2009

From Reinhard Zumkeller, Dec 12 2009: (Start)
A070939(a(n)) = A070939(n) + A033264(n);
A171598 and A171599 give record values and where they occur. (End)

			12 in binary is 1100. Increase each run of 0 by one digit to get 11000, which is 24 in decimal. So a(12) = 24.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A175046, A175048.
Cf. A084471. - Robert G. Wilson v, Dec 11 2009
Cf. A030308, A106151, A007088.

import Data.List (group)
a175047 = foldr (\b v -> 2 * v + b) 0 . concatMap
   (\bs@(b:_) -> if b == 0 then 0 : bs else bs) . group . a030308_row
-- Reinhard Zumkeller, Jul 05 2013

f[n_] := Block[{s = Split@ IntegerDigits[n, 2]}, FromDigits[ Flatten@ Insert[s, {0}, Table[{2 i}, {i, Floor[ Length@s/2]} ]], 2]]; Array[ f, 64] (* Robert G. Wilson v, Dec 11 2009 *)

from re import split
def A175047(n):
  return int(''.join(d+'0' if '0' in d else d for d in split('(0+)|(1+)',bin(n)[2:]) if d != '' and d != None),2) # Chai Wah Wu, Nov 21 2018

a(n) = if n<2 then n else 2*(1 + 0^((n+2) mod 4))*a([n/2]) + n mod 2. - Reinhard Zumkeller, Jan 20 2010

a(2^n) = 2^(n+1). - Chai Wah Wu, Nov 21 2018

Extended by Ray Chandler, Dec 18 2009

a(11) onwards from Robert G. Wilson v and Reinhard Zumkeller, Dec 11 2009

A088698 Replace 1 with 11 in binary representation of n.

Original entry on oeis.org

0, 3, 6, 15, 12, 27, 30, 63, 24, 51, 54, 111, 60, 123, 126, 255, 48, 99, 102, 207, 108, 219, 222, 447, 120, 243, 246, 495, 252, 507, 510, 1023, 96, 195, 198, 399, 204, 411, 414, 831, 216, 435, 438, 879, 444, 891, 894, 1791, 240, 483, 486, 975, 492, 987, 990

Offset: 0

Ralf Stephan, Oct 07 2003

			n=9: 1001 -> 110011 = 51, so a(9) = 51.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Ordered terms plus one are in A048297.
Same sequence sorted into ascending order: A277335, A290258 (without 0).
Cf. A001196, A084471, A048678, A088697.
Main diagonal of A341520, right edge of A341521.

a(n)=if(n<1,0,if(n%2==0,2*a(n/2),4*a((n-1)/2)+3))

def a(n): return int(bin(n)[2:].replace('1', '11'), 2)
print([a(n) for n in range(55)]) # Michael S. Branicky, Feb 20 2021

a(0)=0, a(2n) = 2a(n), a(2n+1) = 4a(n) + 3.

A353654 Numbers whose binary expansion has the same number of trailing 0 bits as other 0 bits.

Original entry on oeis.org

1, 3, 7, 10, 15, 22, 26, 31, 36, 46, 54, 58, 63, 76, 84, 94, 100, 110, 118, 122, 127, 136, 156, 172, 180, 190, 204, 212, 222, 228, 238, 246, 250, 255, 280, 296, 316, 328, 348, 364, 372, 382, 392, 412, 428, 436, 446, 460, 468, 478, 484, 494, 502, 506, 511, 528, 568

Offset: 1

Mikhail Kurkov, Jul 15 2022

Numbers k such that A007814(k) = A086784(k).
To reproduce the sequence through itself, use the following rule: if binary 1xyz is a term then so are 11xyz and 10xyz0 (except for 1 alone where 100 is not a term).
The number of terms with bit length k is equal to Fibonacci(k-1) for k > 1.
Conjecture: 2*A247648(n-1) + 1 with rewrite 1 -> 1, 01 -> 0 applied to binary expansion is the same as a(n) without trailing 0 bits in binary.
Odd terms are positive Mersenne numbers (A000225), because there is no 0 in their binary expansion. - Bernard Schott, Oct 12 2022

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A000120, A000523, A007814, A010056, A025480, A030109, A054429, A060142, A072649, A084471, A086784, A091892, A132665, A133512, A200650, A213911, A232559, A280514, A343152, A348366.
Cf. A356385 (first differences).
Subsequences with same number k of trailing 0 bits and other 0 bits: A000225 (k=0), 2*A190620 (k=1), 4*A357773 (k=2), 8*A360573 (k=3).

N:= 10: # for terms <= 2^N
S:= {1};
for d from 1 to N do
  for k from 0 to d/2-1 do
    B:= combinat:-choose([$k+1..d-2],k);
    S:= S union convert(map(proc(t) local s; 2^d - 2^k - add(2^(s),s=t) end proc,B),set);
od od:
sort(convert(S,list)); # Robert Israel, Sep 21 2023

Join[{1}, Select[Range[2, 600], IntegerExponent[#, 2] == Floor[Log2[# - 1]] - DigitCount[# - 1, 2, 1] &]] (* Amiram Eldar, Jul 16 2022 *)

isok(k) = if (k==1, 1, (logint(k-1, 2)-hammingweight(k-1) == valuation(k, 2))); \\ Michel Marcus, Jul 16 2022

from itertools import islice, count
def A353654_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n & n-1).bit_length()) == bin(n>>m)[2:].count('0'),count(max(startvalue,1)))
A353654_list = list(islice(A353654_gen(),30)) # Chai Wah Wu, Oct 14 2022

a(n) = a(n-1) + A356385(n-1) for n > 1 with a(1) = 1.
Conjectured formulas: (Start)
a(n) = 2^g(n-1)*(h(n-1) + 2^A000523(h(n-1))*(2 - g(n-1))) for n > 2 with a(1) = 1, a(2) = 3 where f(n) = n - A130312(n), g(n) = [n > 2*f(n)] and where h(n) = a(f(n) + 1).
a(n) = 1 + 2^r(n-1) + Sum_{k=1..r(n-1)} (1 - g(s(n-1, k)))*2^(r(n-1) - k) for n > 1 with a(1) = 1 where r(n) = A072649(n) and where s(n, k) = f(s(n, k-1)) for n > 0, k > 1 with s(n, 1) = n.
a(n) = 2*(2 + Sum_{k=1..n-2} 2^(A213911(A280514(k)-1) + 1)) - 2^A200650(n) for n > 1 with a(1) = 1.
A025480(a(n)-1) = A348366(A343152(n-1)) for n > 1.
a(A000045(n)) = 2^(n-1) - 1 for n > 1. (End)

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7

Offset: 0

Alois P. Heinz, Jan 15 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0,        0, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 2,  4,   8,   16,    32,     64,     128,      256, ...
  3, 3,  3,   3,    3,     3,      3,       3,        3, ...
  1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
  3, 5,  9,  17,   33,    65,    129,     257,      513, ...
  3, 6, 12,  24,   48,    96,    192,     384,      768, ...
  7, 7,  7,   7,    7,     7,      7,       7,        7, ...
  1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.
Cf. A000120, A023416.

A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)

A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).

A023416(A(n,k)) = k * A023416(n) for n >= 1.

A084473 Replace 0 with 0000 in binary representation of n.

Original entry on oeis.org

1, 16, 3, 256, 33, 48, 7, 4096, 513, 528, 67, 768, 97, 112, 15, 65536, 8193, 8208, 1027, 8448, 1057, 1072, 135, 12288, 1537, 1552, 195, 1792, 225, 240, 31, 1048576, 131073, 131088, 16387, 131328, 16417, 16432, 2055, 135168, 16897, 16912, 2115, 17152, 2145

Offset: 1

Reinhard Zumkeller, May 27 2003

a(n) = n iff n = 2^k - 1, k>0 (A000225). - Bernard Schott, Dec 18 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..8191

Cf. A000120, A000225, A084471, A084474, A007088, A023416.

Column k=4 of A340666.

a084473 1 = 1
a084473 x = 2 * (if b == 1 then 1 else 8) * a084473 x' + b
            where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 31 2015

a:= n-> Bits[Join](subs(0=[0$4][], Bits[Split](n))):
seq(a(n), n=1..49);  # Alois P. Heinz, Jan 15 2021

a[n_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence@@{0,0,0,0}, 2];
Array[a, 50] (* Jean-François Alcover, Dec 16 2021 *)

def a(n): return int(bin(n)[2:].replace('0', '0000'), 2)
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Jan 15 2021

a(1)=1, a(2*k+1)=2*a(k)+1, a(2*k)=16*a(k).
a(n) = A084471(A084471(n)).
A084474(n) = A007088(a(n));
A023416(a(n)) = A023416(n)*4.
A000120(a(n)) = A000120(n).

A088697 Replace 0 with 10 in binary representation of n.

Original entry on oeis.org

0, 1, 6, 3, 26, 13, 14, 7, 106, 53, 54, 27, 58, 29, 30, 15, 426, 213, 214, 107, 218, 109, 110, 55, 234, 117, 118, 59, 122, 61, 62, 31, 1706, 853, 854, 427, 858, 429, 430, 215, 874, 437, 438, 219, 442, 221, 222, 111, 938, 469, 470, 235, 474, 237

Offset: 0

Ralf Stephan, Oct 07 2003

			n=9: 1001 -> 110101 = 53, so a(9) = 53.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A084471, A048678, A088698.

Join[{0},Table[FromDigits[Flatten[IntegerDigits[n,2]/.(0->{1,0})],2],{n,80}]] (* Harvey P. Dale, Dec 05 2023 *)

a(n)=if(n<1,0,if(n%2==0,4*a(n/2)+2,2*a((n-1)/2)+1))

a(0)=0, a(2n) = 4a(n) + 2, a(2n+1) = 2a(n) + 1.

A229180 Expansion of (chi(-x) * chi(-x^3))^-3 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 6, 16, 33, 60, 118, 210, 354, 612, 1008, 1608, 2583, 4035, 6174, 9448, 14196, 21024, 31054, 45282, 65322, 93884, 133638, 188640, 265225, 370086, 512934, 708136, 971628, 1325724, 1802134, 2437200, 3280452, 4400132, 5876184, 7815288, 10360890, 13683525

Offset: 0

Michael Somos, Sep 30 2013

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

In Verrill (1999) section 2.6, denoted by g as a function of q.

			G.f. = 1 + 3*x + 6*x^2 + 16*x^3 + 33*x^4 + 60*x^5 + 118*x^6 + 210*x^7 + ...
G.f. = q + 3*q^3 + 6*q^5 + 16*q^7 + 33*q^9 + 60*q^11 + 118*q^13 + 210*q^15 + ...

H. Verrill, Some Congruences related to modular forms, Max Planck Institute, 1999.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
H. Verrill, Some Congruences related to modular forms
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A058492, A084471, A217786.

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[x^3, x^6])^3, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)))^3, n))};

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)))^3 in powers of q.
Euler transform of period 6 sequence [3, 0, 6, 0, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = (1/8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058492.
G.f.: t / (1 - 10*t^2 + 9*t^4)^(1/2) where t = the g.f. of A217786.
G.f.: 1 / (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(6*k - 3)))^3.
Convolution inverse of A058492.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

A084472 Write n in binary and replace 0 with 00.

Original entry on oeis.org

1, 100, 11, 10000, 1001, 1100, 111, 1000000, 100001, 100100, 10011, 110000, 11001, 11100, 1111, 100000000, 10000001, 10000100, 1000011, 10010000, 1001001, 1001100, 100111, 11000000, 1100001, 1100100, 110011, 1110000

Offset: 1

Reinhard Zumkeller, May 27 2003

nonn

a(n)=A007088(A084471(n)).

Cf. A084474, A038585.

Table[FromDigits[Flatten[IntegerDigits[n,2]/.(0->{0,0})]],{n,30}] (* Harvey P. Dale, Aug 27 2019 *)

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A048678 Binary expansion of nonnegative integers expanded to "Zeckendorffian format" with rewrite rules 0->0, 1->01.

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A175047 Write n in binary, then increase each run of 0's by one 0. a(n) is the decimal equivalent of the result.

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A088698 Replace 1 with 11 in binary representation of n.

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A353654 Numbers whose binary expansion has the same number of trailing 0 bits as other 0 bits.

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A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A084473 Replace 0 with 0000 in binary representation of n.

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