A000523 -id:A000523 - cp's OEIS frontend

A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

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

Offset: 0

Hieronymus Fischer, May 20 2007

nonn

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=1 (see A130250 for another version). a(n)+1 is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

			a(12)=5, since A001045(5)=11<=12, but A001045(6)=21>12.

G. C. Greubel, Table of n, a(n) for n = 0..10000

For partial sums see A130251.
Other related sequences A130250, A130253, A105348, A001045, A130233, A130241.
Cf. A000523, A078008 (runlengths).

[Floor(Log(3*n+1)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Table[Floor[Log[2, 3*n + 1]], {n, 0, 50}] (* G. C. Greubel, Jan 08 2018 *)

for(n=0, 30, print1(floor(log(3*n+1)/log(2)), ", ")) \\ G. C. Greubel, Jan 08 2018

a(n) = logint(3*n+1, 2); \\ Ruud H.G. van Tol, May 12 2024

def A130249(n): return (3*n+1).bit_length()-1 # Chai Wah Wu, Jun 08 2022

a(n) = floor(log_2(3n+1)).
a(n) = A130250(n+1) - 1 = A130253(n) - 1.
G.f.: 1/(1-x)*(Sum_{k>=1} x^A001045(k)).
a(n) = A000523(3*n+1). - Ruud H.G. van Tol, May 12 2024

A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

Original entry on oeis.org

0, 1, 3, 6, 2, 12, 4, 7, 5, 24, 8, 11, 9, 13, 15, 10, 14, 48, 16, 19, 17, 21, 23, 18, 22, 25, 27, 30, 26, 20, 28, 31, 29, 96, 32, 35, 33, 37, 39, 34, 38, 41, 43, 46, 42, 36, 44, 47, 45, 49, 51, 54, 50, 60, 52, 55, 53, 40, 56, 59, 57, 61, 63, 58, 62, 192, 64, 67, 65, 69, 71, 66, 70, 73, 75, 78, 74, 68, 76, 79, 77, 81

Offset: 0

Antti Karttunen, Feb 12 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A268718.
Cf. A000523, A003188, A003987, A006068, A010060, A066194, A101080, A171977, A268718, A268726, A268727.
Row 1 and column 1 of array A268715 (without the initial zero).
Row 1 of array A268820.
Cf. A092246 (fixed points).
Cf. A268817 ("square" of this permutation).
Cf. A268821 ("shifted square"), A268823 ("shifted cube") and also A268825, A268827 and A268831 ("shifted higher powers").

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := If[n == 0, 0, BitXor @@ Table[Floor[n/2^m], {m, 0, Floor[Log[2, n]]}]]; a[n_] := If[n == 0, 0, A003188[1 + A006068[n-1]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 23 2016 *)

A003188(n) = bitxor(n, floor(n/2));
A006068(n) = if(n<2, n, {my(m = A006068(floor(n/2))); 2*m + (n%2 + m%2)%2});
for(n=0, 100, print1(if(n<1, 0, A003188(1 + A006068(n - 1)))", ")) \\ Indranil Ghosh, Mar 31 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    m = A006068(n//2)
    return 2*m + (n%2 + m%2)%2
def a(n): return 0 if n<1 else A003188(1 + A006068(n - 1))
print([a(n) for n in range(0, 101)]) # Indranil Ghosh, Mar 31 2017

def A268717(n):
    k, m = n-1, n-1>>1
    while m > 0:
        k ^= m
        m >>= 1
    return k+1^ k+1>>1 # Chai Wah Wu, Jun 29 2022

(define (A268717 n) (if (zero? n) n (A003188 (A066194 n))))

a(n) = A003188(A066194(n)) = A003188(1+A006068(n-1)).
Other identities. For all n >= 0:
A101080(n,a(n+1)) = 1. [The Hamming distance between n and a(n+1) is always one.]
A268726(n) = A000523(A003987(n, a(n+1))). [A268726 gives the index of the toggled bit.]
From Alan Michael Gómez Calderón, May 29 2025: (Start)
a(2*n) = (2*n-1) XOR (2-A010060(n-1)) for n >= 1;
a(n) = (A268718(n-1)-1) XOR (A171977(n-1)+1) for n >= 2. (End)

A090996 Number of leading 1's in binary expansion of n.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Feb 29 2004

Mirror of triangle A065120. See example. - Omar E. Pol, Oct 17 2013

a(n) is also the least part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

			In binary : 14=1110 and there are 3 leading 1's, so a(14)=3.
From _Omar E. Pol_, Oct 17 2013: (Start)
Written as an irregular triangle with row lengths A011782 the sequence begins:
0;
1;
1,2;
1,1,2,3;
1,1,1,1,2,2,3,4;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
Right border gives A001477. Row sums give A000225.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..65535 (first 10001 terms from Vincenzo Librandi)
Index entries for sequences related to binary expansion of n

a(n) = A007814(1+A030101(n)).

Cf. A279209, A279210, A360189.

a := proc(n) if type(log[2](n+1), integer) then log[2](n+1) else a(floor((1/2)*n)) end if end proc: seq(a(n), n = 0 .. 200); # Emeric Deutsch, Jul 24 2017
# second Maple program:
b:= proc(n, t) `if`(n=0, t,
      b(iquo(n, 2, 'm'), m*(t+1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..127);  # Alois P. Heinz, Mar 06 2023

Join[{0},Table[Length@First@Split@IntegerDigits[n,2],{n,30}]] (* Birkas Gyorgy, Mar 09 2011 *) (* adapted by Vincenzo Librandi, Dec 23 2016 *)

a(n) = if(n==0, 0); b=binary(n+1); if(hammingweight(b) == 1, #b-1, a(n\2)) \\ David A. Corneth, Jul 24 2017

a(n) = if(n==0, 0); my(b = binary(n), r = #b); for(i=2, #b, if(!b[i], return(i-1))); r \\ David A. Corneth, Jul 24 2017

a(2^k-1)=k; a(A004754(k))=1; a(A004758(k))=2.
a(2^k-1)=k; for any other n, a(n) = a(floor(n/2)).
a(n) = f(n, 0) with f(n, x) = if n < 2 then n + x else f([n/2], (x+1)*(n mod 2)). - Reinhard Zumkeller, Feb 02 2007
Conjecture: a(n) = w(n+1)*(w(n+1)-w(n)+1) - w(2^(w(n+1)+1)-n-1) for n>0, where w(n) = floor(log_2(n)), that is, A000523(n). - Velin Yanev, Dec 21 2016
a(n) = A360189(n-1,floor(log_2(n))). - Alois P. Heinz, Mar 06 2023

Edited and corrected by Franklin T. Adams-Watters, Apr 08 2006

Sequence had accidentally been shifted left by one step, which was corrected and term a(0)=0 added by Antti Karttunen, Jan 01 2007

A151552 G.f.: Product_{k>=1} (1 + x^(2^k-1) + x^(2^k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 4, 3, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 2, 3, 4, 4, 5, 7, 7, 5, 5

Offset: 0

N. J. A. Sloane, May 19 2009, Dec 26 2009

			Written as a triangle:
1;
1;
1,1;
2,2,1,1;
2,2,2,3,4,3,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,1,1;
2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,3,4,4,5,7,7,5,5,...
The rows converge to A151714.

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.

For generating functions of the form Product_{k>=c} (1 + a*x^(2^k-1) + b*x^2^k) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.

Cf. A139250, A151550, A151551, A160573, A151702, A151714, A151553.

G := mul( 1 + x^(2^n-1) + x^(2^n), n=1..20);
wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=proc(n) local t1,k; global wt; t1:=0; for k from 0 to 20 do if n+k mod 2 = 0 then t1:=t1+binomial(wt(n+k),k); fi; od; t1; end;

a[n_] := Sum[If[EvenQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, Floor[Log2[n + 1]]}]; Array[a, 100, 0] (* Amiram Eldar, Jul 29 2023 *)

a(n) = 1 for 0 <= n <= 3; thereafter write n = 2^i + j, with 0 <= j < 2^i, then a(n) = a(j) + a(j+1), except that a(2^(i+1)-2) = a(2^(i+1)-1) = 1.

a(n) = Sum_{k>=0, n+k even} binomial(A000120(n+k),k); the sum may be restricted further to k <= A000523(n+1). - Hagen von Eitzen, May 20 2009 [corrected by Amiram Eldar, Jul 29 2023]

A163356 Inverse permutation to A163355, related to Hilbert's curve in N x N grid.

Original entry on oeis.org

0, 1, 3, 2, 8, 10, 11, 9, 12, 14, 15, 13, 7, 6, 4, 5, 16, 18, 19, 17, 20, 21, 23, 22, 28, 29, 31, 30, 27, 25, 24, 26, 48, 50, 51, 49, 52, 53, 55, 54, 60, 61, 63, 62, 59, 57, 56, 58, 47, 46, 44, 45, 39, 37, 36, 38, 35, 33, 32, 34, 40, 41, 43, 42, 128, 130, 131, 129, 132, 133

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Inverse: A163355.
Second and third "powers": A163906, A163916. See also A059252-A059253.
In range [A000302(n-1)..A024036(n)] of this permutation, the number of cycles is given by A163910, number of fixed points seems to be given by A147600(n-1) (fixed points themselves: A163901). Max. cycle sizes is given by A163911 and LCM's of all cycle sizes by A163912.
Cf. A059252, A059253, A059905, A059906.
Cf. also A302844, A302846, A302781.

A057300(n) = { my(t=1,s=0); while(n>0, if(1==(n%4),n++,if(2==(n%4),n--)); s += (n%4)*t; n >>= 2; t <<= 2); (s); };
A163356(n) = if(!n,n,my(i = (#binary(n)-1)\2, f = 4^i, d = (n\f)%4, r = (n%f)); (((((2+(i%2))^d)%5)-1)*f) + if(3==d,f-1-A163356(r),A057300(A163356(r)))); \\ Antti Karttunen, Apr 14 2018

a(0) = 0, and provided that d=1, 2 or 3, then a((d*(4^i))+r) = (((2+(i mod 2))^d mod 5)-1) * [either A024036(i) - a(r), if d is 3, and A057300(a(r)) in other cases].
From Antti Karttunen, Apr 14 2018: (Start)
A059905(a(n)) = A059253(n).
A059906(a(n)) = A059252(n).
a(n) = A000695(A059253(n)) + 2*A000695(A059252(n)).
(End)

Links to further derived sequences and a nicer Scheme function & formula added by Antti Karttunen, Sep 21 2009

A256290 Numbers which have only digits 4 and 5 in base 10.

Original entry on oeis.org

4, 5, 44, 45, 54, 55, 444, 445, 454, 455, 544, 545, 554, 555, 4444, 4445, 4454, 4455, 4544, 4545, 4554, 4555, 5444, 5445, 5454, 5455, 5544, 5545, 5554, 5555, 44444, 44445, 44454, 44455, 44544, 44545, 44554, 44555, 45444, 45445, 45454, 45455, 45544

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..60000] | Set(IntegerToSequence(n, 10)) subset {5, 4}];

[n: n in [1..100000] | Set(Intseq(n)) subset {4,5}]; // Vincenzo Librandi, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{4,5},n],{n,5}]]

A256290(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*4

def A256290(n): return int(bin(n+1)[3:])+(10**((n+1).bit_length()-1)-1<<2)//9 # Chai Wah Wu, Jul 15 2023

a(n) = A007931(n) + A002277(A000523(n+1)) = A032834(n) + A256077(n) etc.

A256291 Numbers which have only digits 5 and 6 in base 10.

Original entry on oeis.org

5, 6, 55, 56, 65, 66, 555, 556, 565, 566, 655, 656, 665, 666, 5555, 5556, 5565, 5566, 5655, 5656, 5665, 5666, 6555, 6556, 6565, 6566, 6655, 6656, 6665, 6666, 55555, 55556, 55565, 55566, 55655, 55656, 55665, 55666, 56555, 56556

Offset: 1

M. F. Hasler, Mar 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..8190
Index entries for 10-automatic sequences.

Cf. A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A032834 (digits 3 & 4), A256290 (digits 4 & 5), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..60000] | Set(IntegerToSequence(n, 10)) subset {5, 6}];

[n: n in [1..100000] | Set(Intseq(n)) subset {5,6}]; // :Vincenzo Librandi_, Aug 19 2016

Flatten[Table[FromDigits[#,10]&/@Tuples[{5,6},n],{n,5}]]

A256291(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\9*5

a(n) = A007931(n) + A002278(A000523(n+1)) = A256290(n) + A256077(n) etc.

A032834 Numbers with digits 3 and 4 only.

Original entry on oeis.org

3, 4, 33, 34, 43, 44, 333, 334, 343, 344, 433, 434, 443, 444, 3333, 3334, 3343, 3344, 3433, 3434, 3443, 3444, 4333, 4334, 4343, 4344, 4433, 4434, 4443, 4444, 33333, 33334, 33343, 33344, 33433, 33434, 33443, 33444, 34333, 34334

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A032829-A032833 (in other bases), A102659 (Lyndon words in this sequence), A007088 (digits 0 & 1), A007931 (digits 1 & 2), A032810 (digits 2 & 3), A256290 (digits 4 & 5), A256291 (digits 5 & 6), A256292 (digits 6 & 7), A256340 (digits 7 & 8), A256341 (digits 8 & 9).

[n: n in [1..35000] | Set(IntegerToSequence(n, 10)) subset {3, 4}]; // Vincenzo Librandi, May 30 2012

S[1]:= [3,4]:
for d from 2 to 5 do S[d]:= map(t -> (10*t+3,10*t+4), S[d-1]) od:
seq(op(S[d]),d=1..5); # Robert Israel, Apr 03 2017

Flatten[Table[FromDigits[#,10]&/@Tuples[{3,4},n],{n,5}]] (* Vincenzo Librandi, May 30 2012 *)

A032834(n)=vector(#n=binary(n+1)[2..-1],i,10^(#n-i))*n~+10^#n\3 \\ M. F. Hasler, Mar 27 2015

a(n) = A007931(n) + A002276(A000523(n+1)) = A032810(n) + A256077(n) etc. - M. F. Hasler, Mar 27 2015
From Robert Israel, Apr 03 2017: (Start)
a(2*n+1) = 10*a(n)+3.
a(2*n+2) = 10*a(n)+4.
G.f. g(x) satisfies g(x) = 10*(x+x^2)*g(x^2) + x*(3+4*x)/(1-x^2). (End)

Crossrefs added by M. F. Hasler, Mar 27 2015

Name corrected by Robert Israel, Apr 03 2017

A039963 The period-doubling sequence A035263 repeated.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

An example of a d-perfect sequence.
Motzkin numbers mod 2. - Benoit Cloitre, Mar 23 2004
Let {a, b, c, c, a, b, a, b, a, b, c, c, a, b, ...} be the fixed point of the morphism: a -> ab, b -> cc, c -> ab, starting from a; then the sequence is obtained by taking a = 1, b = 1, c = 0. - Philippe Deléham, Mar 28 2004
The asymptotic mean of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021
The Gilbreath transform of floor(log_2(n)) (A000523). - Thomas Scheuerle, Sep 02 2024

Seiichi Manyama, Table of n, a(n) for n = 0..10000
T. Amdeberhan and M. Alekseyev, A moment sequence and Motzkin numbers. Modular coincidence?, MathOverflow, 2021.
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link.
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.

Cf. A000523, A005802, A081706.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Flatten[ Nest[ Function[l, {Flatten[(l /. {a -> {a, b}, b -> {c, c}, c -> {a, b}})]}], {a}, 7] /. {a -> {1}, b -> {1}, c -> {0}}] (* Robert G. Wilson v, Feb 26 2005 *)

A039963(n) = 1 - valuation(n\2+1,2)%2; \\ Max Alekseyev, Oct 23 2021

def A039963(n): return ((m:=(n>>1)+1)&-m).bit_length()&1 # Chai Wah Wu, Jan 09 2023

a(n) = A035263(1+floor(n/2)). - Benoit Cloitre, Mar 23 2004
a(n) = A040039(n) mod 2 = A002212(n+1) mod 2. a(0) = a(1) = 1, for n>=2: a(n) = ( a(n) + Sum_{k=0..n-2} a(k)*a(n-2-k)) mod 2. - Philippe Deléham, Mar 26 2004
a(n) = (A(n+2) - A(n)) mod 2, for A = A019300, A001285, A010060, A010059, A000069, A001969. - Philippe Deléham, Mar 28 2004
a(n) = A001006(n) mod 2. - Christian G. Bower, Jun 12 2005
a(n) = (-1)^n*(A096268(n+1) - A096268(n)). - Johannes W. Meijer, Feb 02 2013
a(n) = 1 - A007814(floor(n/2)+1) mod 2 = A005802(n) mod 2. - Max Alekseyev, Oct 23 2021

More terms from Christian G. Bower, Jun 12 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe and Ralf Stephan, Jul 13 2007

A102572 a(n) = floor(log_4(n)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 23 2006

nonn

Cf. A000523, A062153.

Magma
```
[ Ilog(4,n) : n in [1..150] ];
```

a(n)=#digits(n,4)-1 \\ Twice as fast as a(n)=for(i=0,n,(n>>=2)||return(i)); the naïve code a(n)=log(n)\log(4) works for standard realprecision=28 only up to n=4^47-5 and it is slower by another factor 2. - M. F. Hasler, Mar 11 2015

A102572(n)=logint(n,4) \\ M. F. Hasler, Nov 07 2019

G.f.: (1/(1 - x))*Sum_{k>=1} x^(4^k). - Ilya Gutkovskiy, Jan 08 2017

cp's OEIS Frontend

A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Formula

A268717 Permutation of natural numbers: a(0) = 0, a(n) = A003188(1+A006068(n-1)), where A003188 is binary Gray code and A006068 is its inverse.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Python

Scheme

Formula

A090996 Number of leading 1's in binary expansion of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A151552 G.f.: Product_{k>=1} (1 + x^(2^k-1) + x^(2^k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A163356 Inverse permutation to A163355, related to Hilbert's curve in N x N grid.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A256290 Numbers which have only digits 4 and 5 in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Python

Formula

A256291 Numbers which have only digits 5 and 6 in base 10.

Views