A030308 -id:A030308 - cp's OEIS frontend

A033042 Sums of distinct powers of 5.

0, 1, 5, 6, 25, 26, 30, 31, 125, 126, 130, 131, 150, 151, 155, 156, 625, 626, 630, 631, 650, 651, 655, 656, 750, 751, 755, 756, 775, 776, 780, 781, 3125, 3126, 3130, 3131, 3150, 3151, 3155, 3156, 3250, 3251, 3255, 3256, 3275, 3276, 3280, 3281, 3750, 3751

Offset: 0

Clark Kimberling

Numbers without any base-5 digits larger than 1.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011
Values of k where A008977(k) does not end with 0. - Henry Bottomley, Nov 09 2022

T. D. Noe, Table of n, a(n) for n = 0..1023
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
K. Dilcher and L. Ericksen, Hyperbinary expansions and Stern polynomials, Elec. J. Combin, 22, 2015, #P2.24.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

For generating functions Product_{k>=0} (1 + a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Cf. A000695, A005836, A008977, A010060, A033043-A033052.
Row 5 of array A104257.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 5
    end
r end; [a(n) for n in 0:49] |> println # Peter Luschny, Jan 03 2021

a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*5 od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 16 2013

t = Table[FromDigits[RealDigits[n, 2], 5], {n, 1, 100}]
(* Clark Kimberling, Aug 02 2012 *)
FromDigits[#,5]&/@Tuples[{0,1},7] (* Harvey P. Dale, May 22 2018 *)

a(n) = subst(Pol(binary(n)), 'x, 5);
vector(50, i, a(i-1))  \\ Gheorghe Coserea, Sep 15 2015

a(n)=fromdigits(binary(n),5) \\ Charles R Greathouse IV, Jan 11 2017

def A033042(n): return int(bin(n)[2:],5) # Chai Wah Wu, Oct 30 2024

a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
Numbers j such that the coefficient of x^j is > 0 in Product_{k>=0} (1 + x^(5^k)). - Benoit Cloitre, Jul 29 2003
a(n) = A097251(n)/4.
a(2n) = 5*a(n), a(2n+1) = a(2n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*5^k. - Philippe Deléham, Oct 17 2011
liminf a(n)/n^(log(5)/log(2)) = 1/4 and limsup a(n)/n^(log(5)/log(2)) = 1. - Gheorghe Coserea, Sep 15 2015
G.f.: (1/(1 - x))*Sum_{k>=0} 5^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extended by Ray Chandler, Aug 03 2004

A047218 Numbers that are congruent to {0, 3} mod 5.

Original entry on oeis.org

0, 3, 5, 8, 10, 13, 15, 18, 20, 23, 25, 28, 30, 33, 35, 38, 40, 43, 45, 48, 50, 53, 55, 58, 60, 63, 65, 68, 70, 73, 75, 78, 80, 83, 85, 88, 90, 93, 95, 98, 100, 103, 105, 108, 110, 113, 115, 118, 120, 123, 125, 128, 130, 133, 135, 138, 140, 143, 145, 148

Offset: 1

N. J. A. Sloane

Multiples of 5 interleaved with 2 less than multiples of 5. - Wesley Ivan Hurt, Oct 19 2013

Numbers k such that k^2/5 + k*(k + 1)/10 = k*(3*k + 1)/10 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..10001
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A047211, A047212, A047216.

seq(floor(5*k/2)-2, k=1..100); # Wesley Ivan Hurt, Sep 27 2013

Select[Range[0, 200], MemberQ[{0, 3}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)
Table[Floor[5 n/2] - 2, {n,100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
With[{c5=5*Range[0,30]},Riffle[c5,c5+3]] (* or *) LinearRecurrence[{1,1,-1},{0,3,5},60] (* Harvey P. Dale, Apr 02 2017 *)

forstep(n=0,200,[3,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = 2*n - 5 + ceiling(n/2). - Jesus De Loera (deloera(AT)math.ucdavis.edu)
a(n) = 5*n - a(n-1) - 7 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010
From Bruno Berselli, Jun 28 2011: (Start)
G.f.: (2*x + 3)*x^2/((x + 1)*(x - 1)^2).
a(n) = (10*n + (-1)^n - 9)/4.
a(n) = a(n-1) + a(n-2) - a(n-3).  (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=3 and b(k)=A020714(k-1)=5*2^(k-1) for k>0. - Philippe Deléham, Oct 17 2011
a(n) = n + ceiling(3*(n-1)/2) - 1. - Arkadiusz Wesolowski, Sep 18 2012
a(n) = floor(5*n/2)-2 = 3*n - 3 - floor((n-1)/2). - Wesley Ivan Hurt, Oct 14 2013
a(n+1) = n + (n + (n + (n mod 2))/2). - Wesley Ivan Hurt, Oct 19 2013
Sum_{n>=2} (-1)^n/a(n) = log(5)/4 - sqrt(5)*log(phi)/10 - sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: 2 + ((5*x - 9/2)*exp(x) + (1/2)*exp(-x))/2. - David Lovler, Aug 22 2022

A076478 The binary Champernowne sequence: concatenate binary vectors of lengths 1, 2, 3, ... in numerical order.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Nov 10 2002

Can also be seen as triangle where row n contains all binary vectors of length n+1. - Reinhard Zumkeller, Aug 18 2015
From Clark Kimberling, Jul 18 2021: (Start)
In the following list, W represents the sequence of words w(n) represented by A076478. The list includes five partitions and two self-inverse permutations of the positive integers.
length of w(n): A000523
positions in W of words w(n) such that # 0's = # 1's: A258410;
positions in W of words w(n) such that # 0's < # 1's: A346299;
positions in W of words w(n) such that # 0's > # 1's: A346300;
positions in W of words w(n) that end with 0: A005498;
positions in W of words w(n) that end with 1: A005843;
positions in W of words w(n) such that first digit = last digit: A346301;
positions in W of words w(n) such that first digit != last digit: A346302;
positions in W of words w(n) such that 1st digit = 0 and last digit 0: A171757;
positions in W of words w(n) such that 1st digit = 0 and last digit 1: A346303;
positions in W of words w(n) such that 1st digit = 1 and last digit 0: A346304;
positions in W of words w(n) such that 1st digit = 1 and last digit 1: A346305;
position in W of n-th positive integer (base 2):  A206332;
positions in W of binary complement of w(n):  A346306;
sum of digits in w(n): A048881;
number of runs in w(n): A346307;
positions in W of palindromes: A346308;
positions in W of words such that #0's - #1's is odd: A346309;
positions in W of words such that #0's - #1's is even: A346310;
positions in W of the reversal of the n-th word in W: A081241. (End)

			0,
1,
0,0,
0,1,
1,0,
1,1,
0,0,0,
0,0,1,
0,1,0,
0,1,1,
1,0,0,
1,0,1,
...

Bodil Branner, Dynamics, Chap. IV.14 of The Princeton Companion to Mathematics, ed. T. Gowers, p. 499.
K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Math. Assoc. America, 2002, p. 72.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michael Barnsley and Andrew Vince, Self-similar polygonal tiling, The American Mathematical Monthly 124.10 (2017): 905-921. See page 917.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

Cf. A007931, A030308, A053645.

Cf. A341256, A341258, A341334, A342753, A342910.

import Data.List (unfoldr)
a076478 n = a076478_list !! n
a076478_list = concat $ tail $ map (tail . reverse . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2 )) [1..]
-- Reinhard Zumkeller, Feb 08 2012

a076478_row n = a076478_tabf !! n :: [[Int]]
a076478_tabf = tail $ iterate (\bs -> map (0 :) bs ++ map (1 :) bs) [[]]
a076478_list' = concat $ concat a076478_tabf
-- Reinhard Zumkeller, Aug 18 2015

d[n_] := Rest@IntegerDigits[n + 1, 2] + 1; -1 + Flatten[Array[d, 50]] (* Clark Kimberling, Feb 07 2012 *)
z = 1000;
t1 = Table[Tuples[{0, 1}, n], {n, 1, 10}];
"All binary words, lexicographic order:"
tt = Flatten[t1, 1]; (* all binary words, lexicographic order *)
"All binary words, flattened:"
Flatten[tt];
w[n_] := tt[[n]];
"List tt of all binary words:"
tt = Table[w[n], {n, 1, z}]; (*  all the binary words *)
u1 = Flatten[tt]; (* words, concatenated, A076478, binary Champernowne sequence *)
u2 = Map[Length, tt];
"Positions of 0^n:"
Flatten[Position[Map[Union, tt], {0}]]
"Positions of 1^n:"
Flatten[Position[Map[Union, tt], {1}]]
"Positions of words in which #0's = #1's:"  (* A258410 *)
"This and the next two sequences partition N."
u3 = Select[Range[Length[tt]], Count[tt[[#]], 0] == Count[tt[[#]], 1] &]
"Positions of words in which #0's < #1's:"  (* A346299 *)
u4 = Select[Range[Length[tt]], Count[tt[[#]], 0] < Count[tt[[#]], 1] &]
"Positions of words in which #0's > #1's:"  (* A346300 *)
u5 = Select[Range[Length[tt]], Count[tt[[#]], 0] > Count[tt[[#]], 1] &]
"Positions of words ending with 0:" (* A005498 *)
u6 = Select[Range[Length[tt]], Last[tt[[#]]] == 0 &]
"Positions of words ending with 1:" (* A005843 *)
u7 = Select[Range[Length[tt]], Last[tt[[#]]] == 1 &]
"Positions of words starting and ending with same digit:" (* A346301 *)
u8 = Select[Range[Length[tt]], First[tt[[#]]] == Last[tt[[#]]] &]
"Positions of words starting and ending with opposite digits:" (* A346302  *)
u9 = Select[Range[Length[tt]], First[tt[[#]]] != Last[tt[[#]]] &]
"Positions of words starting with 0 and ending with 0:" (* A346303 *)
"This and the next three sequences partition N."
u10 = Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 0 and ending with 1:" (* A171757 *)
u11 = Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &]
"Positions of words starting with 1 and ending with 0:" (* A346304 *)
u12 = Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 1 and ending with 1:" (* A346305 *)
u13 = Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &]
"Position of n-th positive integer (base 2) in tt:"
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
u14 = Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]] (* A206332 *)
"Position of binary complement of w(n):"
u15 = comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]] (* A346306 *)
"Sum of digits of w(n):"
u16 = Table[Total[w[n]], {n, 1, 100}] (* A048881 *)
"Number of runs in w(n):"
u17 = Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]] (* A346307 *)
"Palindromes:"
Select[tt, # == Reverse[#] &]
"Positions of palindromes:"
u18 = Select[Range[Length[tt]], tt[[#]] == Reverse[tt[[#]]] &] (* A346308 *)
"Positions of words in which #0's - #1's is odd:"
u19 = Select[Range[Length[tt]], OddQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A346309 *)
"Positions of words in which #0's - #1's is even:"
u20 = Select[Range[Length[tt]], EvenQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A346310 *)
"Position of the reversal of the n-th word:"  (* A081241 *)
u21 = Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]]
(* Clark Kimberling, Jul 18 2011 *)

{m=5; for(d=1,m, for(k=0,2^d-1,v=binary(k); while(matsize(v)[2]

listn(n)= my(a=List(), i=0, s=0); while(s<=n, listput(~a, binary(i++)[^1]); s+=#a[#a]); concat(a)[1..n+1]; \\ Ruud H.G. van Tol, Mar 17 2025

from itertools import count, product
def agen():
    for digits in count(1):
        for b in product([0, 1], repeat=digits):
            yield from b
g = agen()
print([next(g) for n in range(105)]) # Michael S. Branicky, Jul 18 2021

To get the m-th binary vector, write m+1 in base 2 and remove the initial 1. - Clark Kimberling, Feb 07 2010

Extended by Klaus Brockhaus, Nov 11 2002

A047215 Numbers that are congruent to {0, 2} mod 5.

Original entry on oeis.org

0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 150, 152, 155, 157

Offset: 0

N. J. A. Sloane

Number of partitions of 5n into exactly 2 parts. - Colin Barker, Mar 23 2015

Numbers k such that k^2/5 + k*(k + 1)/5 = k*(2*k + 1)/5 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

David Lovler, Table of n, a(n) for n = 0..1000
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Different from A038126.
Cf. A001622, A010693, A030308, A084215
Cf. A249547 (partial sums), A010693 (1st differences).

[(5*n - (n mod 2))/2: n in [1..100]]; // G. C. Greubel, Jun 23 2024

Table[Floor[5 n/2], {n, 0, 100}] (* or *) LinearRecurrence[{1, 1, -1}, {0, 2, 5},101] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)

a(n)=5*n\2 \\ Charles R Greathouse IV, Oct 17 2011

[int(5*n//2) for n in range(1,101)] # G. C. Greubel, Jun 23 2024

a(n) = floor(5*n/2).
From R. J. Mathar, Sep 23 2008: (Start)
G.f.: x*(2 + 3*x)/((1 + x)*(1 - x)^2).
a(n) = 5*n/2 + ((-1)^n-1)/4.
a(n+1) - a(n) = A010693(n+1). (End)
a(n) = 5*n - a(n-1) - 8 with a(1)=0. - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A084215(k+1). - Philippe Deléham, Oct 17 2011
a(n) = 2*n + floor(n/2). - Arkadiusz Wesolowski, Sep 19 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = log(5)/4 - sqrt(5)*log(phi)/10 + sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: (5*x*exp(x) - sinh(x))/2. - David Lovler, Aug 22 2022

A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

Original entry on oeis.org

1, 2, 3, 5, 9, 19, 39, 79, 157, 310, 621, 1246, 2491, 4980, 9958, 19924, 39844, 79672, 159365, 318736, 637457, 1274916, 2549816, 5099651, 10199363, 20398663, 40797299, 81594571, 163189087, 326378438, 652756861, 1305513511, 2611026987, 5222053970, 10444108084

Offset: 0

Labos Elemer, Nov 14 2002

nonn

Number of squarefree numbers with binary expansion of length n, or with n bits. The sum of these numbers is given by A373123. - Gus Wiseman, Jun 02 2024

			For n=4: among the 16 numbers of {16, ..., 31}, nine are squarefree [17, 19, 21, 22, 23, 26, 29, 30, 31], so a(4) = 9.

Amiram Eldar, Table of n, a(n) for n = 0..63 (calculated from the b-file at A143658)

Cf. A077641, A077642.
Partial sums (except first term) are A143658.
Run-lengths of A372475.
The minimum is A372683, delta A373125, indices A372540.
The maximum is A372889 (except at n=1), delta A373126, indices A143658.
Row-sums are A373123.
A005117 lists squarefree numbers, first differences A076259.
A053797 gives nonempty lengths of exclusive gaps between squarefree numbers.
A029837 counts bits, row-lengths of A030190 and A030308.
For primes between powers of 2:
- sum A293697
- length A036378 or A162145
- min A104080 or A014210, delta A092131, indices A372684
- max A014234, delta A013603, indices A007053
For squarefree numbers between primes:
- sum A373197
- length A373198 = A061398 - 1
- min A000040
- max A112925 (delta A240473), opposite A112926 (delta A240474)
Cf. A010036, A029931, A035100, A049093-A049096, A372473 (firsts of A372472), A372541 (firsts of A372433).

Table[Apply[Plus, Table[Abs[MoebiusMu[2^w+j]], {j, 0, 2^w-1}]], {w, 0, 15}]
(* second program *)
Length/@Split[IntegerLength[Select[Range[10000],SquareFreeQ],2]]//Most (* Gus Wiseman, Jun 02 2024 *)

{ a(n) = sum(m=1,sqrtint(2^(n+1)-1), moebius(m) * ((2^(n+1)-1)\m^2 - (2^n-1)\m^2) ) } \\ Max Alekseyev, Oct 18 2008

a(n) = Sum_{j=0..-1+2^n} abs(mu(2^n + j)).

a(n)/2^n approaches 1/zeta(2), so limiting sequence is floor(2^n/zeta(2)), n >= 0. - Wouter Meeussen, May 25 2003

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 12 2003
More terms from Wouter Meeussen, May 25 2003
a(25)-a(32) from Max Alekseyev, Oct 18 2008
a(33)-a(34) from Amiram Eldar, Jul 17 2024

A133457 Irregular triangle read by rows: row n gives exponents in expression for n as a sum of powers of 2.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1, 2, 3, 0, 3, 1, 3, 0, 1, 3, 2, 3, 0, 2, 3, 1, 2, 3, 0, 1, 2, 3, 4, 0, 4, 1, 4, 0, 1, 4, 2, 4, 0, 2, 4, 1, 2, 4, 0, 1, 2, 4, 3, 4, 0, 3, 4, 1, 3, 4, 0, 1, 3, 4, 2, 3, 4, 0, 2, 3, 4, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 0, 5, 1, 5, 0, 1, 5, 2, 5, 0, 2, 5, 1, 2, 5, 0, 1, 2, 5, 3, 5, 0, 3, 5

Offset: 1

Masahiko Shin, Nov 27 2007

This sequence contains every increasing finite sequence. For example, the finite sequence {0,2,3,5} arises from n = 45.
Essentially A030308(n,k)*k, then entries removed where A030308(n,k)=0. - R. J. Mathar, Nov 30 2007
In the corresponding irregular triangle {a(n)+1}, the m-th row gives all positive integer roots m_i of polynomial {m,k}. - see link [Shevelev]; see also A264613. - Vladimir Shevelev, Dec 13 2015

			1 = 2^0.
2 = 2^1.
3 = 2^0 + 2^1.
4 = 2^2.
5 = 2^0 + 2^2.
etc. and reading the exponents gives the rows of the triangle.

Reinhard Zumkeller, Rows n = 1..1024 of triangle, flattened
Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. (See Section 3, Theorem 21 and Section 8, Theorem 50)

Cf. A003071, A030308, A048793, A067255, A264613.

Cf. A073642 (row sums), A272011 (rows reversed).

a133457 n k = a133457_tabf !! (n-1) !! n
a133457_row n = a133457_tabf !! (n-1)
a133457_tabf = map (fst . unzip . filter ((> 0) . snd) . zip [0..]) $
                   tail a030308_tabf
-- Reinhard Zumkeller, Oct 28 2013, Feb 06 2013

A133457 := proc(n) local a,bdigs,i ; a := [] ; bdigs := convert(n,base,2) ; for i from 1 to nops(bdigs) do if op(i,bdigs) <> 0 then a := [op(a),i-1] ; fi ; od: a ; end: seq(op(A133457(n)),n=1..80) ; # R. J. Mathar, Nov 30 2007

Array[Join @@ Position[#, 1] - 1 &@ Reverse@ IntegerDigits[#, 2] &, 41] // Flatten (* Michael De Vlieger, Oct 08 2017 *)

a(n) = A048793(n) - 1.

More terms from R. J. Mathar, Nov 30 2007

A038554 Derivative of n: write n in binary, replace each pair of adjacent bits with their mod 2 sum (a(0)=a(1)=0 by convention). Also n XOR (n shift 1).

Original entry on oeis.org

0, 0, 1, 0, 2, 3, 1, 0, 4, 5, 7, 6, 2, 3, 1, 0, 8, 9, 11, 10, 14, 15, 13, 12, 4, 5, 7, 6, 2, 3, 1, 0, 16, 17, 19, 18, 22, 23, 21, 20, 28, 29, 31, 30, 26, 27, 25, 24, 8, 9, 11, 10, 14, 15, 13, 12, 4, 5, 7, 6, 2, 3, 1, 0, 32, 33, 35, 34, 38, 39, 37, 36, 44, 45, 47, 46, 42, 43, 41, 40, 56, 57

Offset: 0

N. J. A. Sloane

From Antti Karttunen: this is also a version of A003188: a(n) = A003188(n) - 2^floor(log_2(A003188(n))), that is, the corresponding Gray code expansion, but with highest 1-bit turned off. Also a(n) = A003188(n) - 2^floor(log_2(n)).
From John W. Layman: {a(n)} is a self-similar sequence under Kimberling's "upper-trimming" operation.
a(A000225(n)) = 0; a(A062289(n)) > 0; a(A038558(n)) = n. - Reinhard Zumkeller, Mar 06 2013

			If n = 18 = 10010_2, derivative is (1+0)(0+0)(0+1)(1+0) = 1011_2, so a(18)=11.

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

T. D. Noe, Table of n, a(n) for n = 0..4096
Clark Kimberling, Fractal sequences.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
J. W. Layman, View the fractal-like graph of a(n) vs. n.
Ralf Stephan, Some divide-and-conquer sequences ....
Ralf Stephan, Table of generating functions.

Cf. A038570, A038571. See A003415 for another definition of the derivative of a number.
Cf. A038556 (rotates n instead of shifting).
Cf. A000035.
Cf. A030308.

import Data.Bits (xor)
a038554 n = foldr (\d v -> v * 2 + d) 0 $ zipWith xor bs $ tail bs
   where bs = a030308_row n
-- Reinhard Zumkeller, May 26 2013, Mar 06 2013

A038554 := proc(n) local i,b,ans; ans := 0; b := convert(n,base,2); for i to nops(b)-1 do ans := ans+((b[ i ]+b[ i+1 ]) mod 2)*2^(i-1); od; RETURN(ans); end; [ seq(A038554(i),i=0..100) ];

a[0] = a[1] = 0; a[n_ /; Mod[n, 4] == 0] := a[n] = 2*a[n/2]; a[n_ /; Mod[n, 4] == 1] := a[n] =  2*a[(n-1)/2] + 1; a[n_ /; Mod[n, 4] == 2] := a[n] = 2*a[n/2] + 1; a[n_ /; Mod[n, 4] == 3] := a[n] = 2*a[(n-1)/2]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Jul 13 2012, after Ralf Stephan *)
Table[FromDigits[Mod[Total[#],2]&/@Partition[IntegerDigits[n,2],2,1],2],{n,0,90}] (* Harvey P. Dale, Oct 27 2015 *)

a003188(n)=bitxor(n, n>>1);
a(n)=if(n<2, 0, a003188(n) - 2^logint(a003188(n), 2)); \\ Indranil Ghosh, Apr 26 2017

import math
def a003188(n): return n^(n>>1)
def a(n): return 0 if n<2 else a003188(n) - 2**int(math.floor(math.log(a003188(n), 2))) # Indranil Ghosh, Apr 26 2017

If 2*2^k <= n < 3*2^k then a(n) = 2^k + a(2^(k+2)-n-1); if 3*2^k <= n < 4*2^k then a(n) = a(n-2^(k+1)). - Henry Bottomley, May 11 2000
G.f.: (1/(1-x)) * Sum_{k>=0} 2^k*(t^4-t^3+t^2)/(1+t^2), t=x^2^k. - Ralf Stephan, Sep 10 2003
a(0)=0, a(2n) = 2*a(n) + [n odd], a(2n+1) = 2*a(n) + [n>0 even]. - Ralf Stephan, Oct 20 2003
a(0) = a(1) = 0, a(4n) = 2*a(2n), a(4n+2) = 2*a(2n+1)+1, a(4n+1) = 2*a(2n)+1, a(4n+3) = 2*a(2n+1). Proof by Nikolaus Meyberg following a conjecture by Ralf Stephan.

More terms from Erich Friedman

A245563 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with low-order bits.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 10 2014

A formula for A071053(n) depends on this table.

			Here are the run lengths for the numbers 0 through 21:
0, []
1, [1]
2, [1]
3, [2]
4, [1]
5, [1, 1]
6, [2]
7, [3]
8, [1]
9, [1, 1]
10, [1, 1]
11, [2, 1]
12, [2]
13, [1, 2]
14, [3]
15, [4]
16, [1]
17, [1, 1]
18, [1, 1]
19, [2, 1]
20, [1, 1]
21, [1, 1, 1]

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Index entries for sequences related to binary expansion of n

Row sums = A000120 (the binary weight).
Cf. A245562, A071053.
Cf. A030308, A227736.
Row lengths are A069010.
The version for prime indices (instead of binary indices) is A124010.
Numbers with distinct run-lengths are A328592.
Numbers with equal run-lengths are A164707.
Cf. A003714, A014081, A328166, A328594, A328595, A328596.

import Data.List (group)
a245563 n k = a245563_tabf !! n !! k
a245563_row n = a245563_tabf !! n
a245563_tabf = [0] : map
   (map length . (filter ((== 1) . head)) . group) (tail a030308_tabf)
-- Reinhard Zumkeller, Aug 10 2014

for n from 0 to 128 do
lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
elif out1 = 0 and t1[i] = 1 then c:=c+1;
elif out1 = 1 and t1[i] = 0 then c:=c;
elif out1 = 0 and t1[i] = 0 then lis:=[op(lis),c]; out1:=1; c:=0;
fi;
if i = L1 and c>0 then lis:=[op(lis),c]; fi;
od:
lprint(n,lis);
od:

Join@@Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}] (* Gus Wiseman, Nov 03 2019 *)

from re import split
A245563_list = [0]
for n in range(1,100):
    A245563_list.extend(len(d) for d in split('0+',bin(n)[:1:-1]) if d != '')
# Chai Wah Wu, Sep 07 2014

A165413 a(n) is the number of distinct lengths of runs in the binary representation of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Sep 17 2009

Least k whose value is n: 1, 4, 35, 536, 16775, 1060976, ..., = A165933. - Robert G. Wilson v, Sep 30 2009

			92 in binary is 1011100. There is a run of one 1, followed by a run of one 0, then a run of three 1's, then finally a run of two 0's. The run lengths are therefore (1,1,3,2). The distinct values of these run lengths are (1,3,2). Since there are 3 distinct values, then a(92) = 3.

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

Cf. A140690 (locations of 1's), A165933 (locations of new highs).
Cf. A005811, A165414.
Cf. A030308, A044813.

import Data.List (group, nub)
a165413 = length . nub . map length . group . a030308_row
-- Reinhard Zumkeller, Mar 02 2013

f[n_] := Length@ Union@ Map[ Length, Split@ IntegerDigits[n, 2]]; Array[f, 105] (* Robert G. Wilson v, Sep 30 2009 *)

binruns(n) = {
  if (n == 0, return([1, 0]));
  my(bag = List(), v=0);
  while(n != 0,
        v = valuation(n,2); listput(bag, v); n >>= v; n++;
        v = valuation(n,2); listput(bag, v); n >>= v; n--);
  return(Vec(bag));
};
a(n) = #Set(select(k->k, binruns(n)));
vector(105, i, a(i))  \\ Gheorghe Coserea, Sep 17 2015

from itertools import groupby
def a(n): return len(set([len(list(g)) for k, g in groupby(bin(n)[2:])]))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jan 04 2021

a(n) = 1 for n in A140690. - Robert G. Wilson v, Sep 30 2009

More terms from Robert G. Wilson v, Sep 30 2009

A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 0, 1, 1, 1, 1, 1, 2

Offset: 0

Clark Kimberling

If columns are numbered starting with k=0, then T(n,k) contains the coefficient of 6^k in n's base-6 expansion. - M. F. Hasler, Jul 21 2013

R. J. Mathar, Table of n, a(n) for n = 0..5950

See A030548 for a quite complete list of crossreferences.
Cf. A030568 - A030573 for positions of a given digit.
Cf. A030575 - A030580 for run lengths, A030581 - A030585 for more.
Row sums (same as those of A030548) are in A053827.
Cf. A030308, A030341, A030386, A031235, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,6]],{n,0,50}]] (* Harvey P. Dale, Sep 27 2015 *)

A030567(n,k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\6^k%6 \\ Assuming that columns start with k=0, cf. comment. TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

Initial 0 and better name from Philippe Deléham, Oct 20 2011

Edited and crossrefs added by M. F. Hasler, Jul 21 2013

cp's OEIS Frontend

A033042 Sums of distinct powers of 5.

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A047218 Numbers that are congruent to {0, 3} mod 5.

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A076478 The binary Champernowne sequence: concatenate binary vectors of lengths 1, 2, 3, ... in numerical order.

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A047215 Numbers that are congruent to {0, 2} mod 5.

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A077643 Number of squarefree integers in closed interval [2^n, -1 + 2*2^n], i.e., among 2^n consecutive numbers beginning with 2^n.

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A133457 Irregular triangle read by rows: row n gives exponents in expression for n as a sum of powers of 2.

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