A055010 -id:A055010 - cp's OEIS frontend

A086784 Number of non-trailing zeros in binary representation of n.

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

Offset: 0

Reinhard Zumkeller, Aug 03 2003

a(n) is also the number of parts smaller than the largest 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

			a(34) = 3; indeed the binary representation of 34 is 100010, having 3 non-trailing zeros. - _Emeric Deutsch_ Jul 24 2017

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Binary Carry Sequence
Index entries for sequences related to binary expansion of n

Cf. A007088.

a := proc (n) local b, c: b := proc (n) if `mod`(n, 2) = 0 then 1+b((1/2)*n) else 0 end if end proc: c := proc (n) if n = 0 then 2 elif n = 1 then 0 elif `mod`(n, 2) = 0 then 1+c((1/2)*n) else c((1/2)*n-1/2) end if end proc: if n = 0 then 0 else c(n)-b(n) end if end proc: seq(a(n), n = 0 .. 101); # b and c are the Maple programs for A007814 and A023416, respectively. - Emeric Deutsch Jul 24 2017

A086784[n_] := If[n == 0, 0, DigitCount[n, 2, 0] - IntegerExponent[n, 2]];
Array[A086784, 100, 0] (* Paolo Xausa, Oct 01 2024 *)

a(n)=if(n==0,0,exponent(n)+1-hammingweight(n)-valuation(n,2)); \\ Antoine Mathys, Nov 20 2024

def A086784(n): return bin(n>>(~n & n-1).bit_length())[2:].count('0') if n else 0 # Chai Wah Wu, Oct 14 2022

a(n) = A023416(n) - A007814(n) for n>0.
a(2^n) = a(A000079(n)) = 0; a(2^n - 1) = a(A000225(n)) = 0;
a(2^n + 1) = a(A000051(n)) = n - 1;
a(3*2^n - 1) = a(A055010(n)) = 1 for n>0;
a(2^n - 3) = a(A036563(n)) = 1, for n>2;
a((4^n - 1)/3) = a(A002450(n)) = n.
a(n) = if n mod 4 = 1 then a(floor(n/4)) + A007814(floor(n/2)) else a(floor(n/2)); a(0) = a(1) = 0.

A182461 a(n) = 3a(n-1) - 2a(n-2) with a(0)=16 and a(1)=40.

Original entry on oeis.org

16, 40, 88, 184, 376, 760, 1528, 3064, 6136, 12280, 24568, 49144, 98296, 196600, 393208, 786424, 1572856, 3145720, 6291448, 12582904, 25165816, 50331640, 100663288, 201326584, 402653176, 805306360, 1610612728, 3221225464, 6442450936, 12884901880

Offset: 0

Odimar Fabeny, Apr 30 2012

Number of vertices into building blocks of 3d objects with 4 vertices.

			a(0) = 4+8+4;
a(1) = 4+8+16+8+4;
a(2) = 4+8+16+32+16+8+4;
a(3) = 4+8+16+32+64+32+16+8+4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -2).

Cf. A000045, A028399, A038578, A089143, A173033, A182462, A182464, A182465, A182466, A182467.

CoefficientList[Series[-((8 (x - 2))/(2 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 02 2014 *)

a(n) = a(n-1)*2 + 8.
G.f.: 16 + 40*x + 88*x^2 + 184*x^3 + 376*x^4 + 760*x^5 + 1528*x^6 + ...
a(n) = 8 * A055010(n+1). [Joerg Arndt, Jun 01 2014]
G.f.: -((8*(x - 2))/(2*x^2 - 3*x + 1)). - Vincenzo Librandi, Jun 02 2014

A275726 A275725-polynomials evaluated at x=2: a(n) = A048675(A275725(n)).

Original entry on oeis.org

1, 2, 5, 3, 4, 3, 11, 10, 7, 4, 5, 4, 9, 7, 7, 4, 6, 4, 8, 7, 6, 4, 5, 4, 23, 22, 21, 19, 20, 19, 15, 14, 9, 5, 6, 5, 11, 8, 9, 5, 8, 5, 9, 8, 7, 5, 6, 5, 19, 18, 15, 12, 13, 12, 15, 14, 9, 5, 6, 5, 13, 11, 9, 5, 7, 5, 12, 11, 8, 5, 6, 5, 17, 15, 15, 12, 14, 12, 13, 11, 9, 5, 7, 5, 11, 8, 9, 5, 8, 5, 10, 8, 8, 5, 7, 5

Offset: 0

Antti Karttunen, Aug 09 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..40319
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A000142, A048675, A055010, A275725.

Cf. A275833 (indices of odd terms), A275834 (of even terms).

(define (A275726 n) (A048675 (A275725 n)))

a(n) = A048675(A275725(n)).
Other identities:
For n >= 1, a(n!) = A055010(n).

A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Robert Price, Jul 19 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Essentially the same as A153893, A083329, A055010, A052940, A266550.

Cf. A290111, A290112, A290113, .

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

For n>1, a(n) = 3*2^(n-1)-1.
a(n) = A266550(n+2) for n > 1. - Georg Fischer, Oct 30 2018
a(n) = 2*a(n-1) + 1 for n=1 and n>=3. - Gennady Eremin, Aug 26 2023
From Chai Wah Wu, Apr 02 2024: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 3.
G.f.: (2*x^3 - 2*x^2 + 1)/((x - 1)*(2*x - 1)). (End)

A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 11, 6, 31, 16, 23, 12, 9, 63, 32, 47, 24, 19, 10, 127, 64, 95, 48, 39, 20, 13, 255, 128, 191, 96, 79, 40, 27, 14, 511, 256, 383, 192, 159, 80, 55, 28, 17, 1023, 512, 767, 384, 319, 160, 111, 56, 35, 18, 2047, 1024, 1535, 768, 639

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000225 (-1+2^n)
Row 2:  A000079 (2^n)
Row 3:  A055010
Row 4:  3*A000079
Row 5:  A153894
Row 6:  5*A000079
Row 7:  A086224
Row 8:  A005009
Row 9:  A052996
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
This sequence is a permutation of the natural numbers.  - L. Edson Jeffery, Aug 13 2014

			Northwest corner:
1...3...7....15...31
2...4...8....16...32
5...11..23...47...95
6...12..24...48...96
9...19..39...79...159

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191665, A191426.

Cf. A241957.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A014601(n+2): (4+4k,5+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191664 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664  *)
(* Clark Kimberling, Jun 11 2011 *)
Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* L. Edson Jeffery, Aug 13 2014 *)

A196168 In binary representation of n: replace each 0 with 1, and each 1 with 10.

Original entry on oeis.org

1, 2, 5, 10, 11, 22, 21, 42, 23, 46, 45, 90, 43, 86, 85, 170, 47, 94, 93, 186, 91, 182, 181, 362, 87, 174, 173, 346, 171, 342, 341, 682, 95, 190, 189, 378, 187, 374, 373, 746, 183, 366, 365, 730, 363, 726, 725, 1450, 175, 350, 349, 698, 347, 694, 693, 1386

Offset: 0

Reinhard Zumkeller, Oct 28 2011

nonn

All terms are numbers with no two adjacent zeros in binary representation, cf. A003754;
a(odd) = even and a(even) = odd;
A023416(a(n)) <= A000120(a(n)), equality iff n = 2^k - 1 for k > 0;
A055010(n+1) = A196168(A000079(n));
A000120(a(n)) = A070939(n);
A023416(a(n)) = A000120(n);
A070939(a(n)) = A070939(n) + A000120(n).

			n =  7 ->  111 ->  101010 ->  a(7) = 42;
n =  8 -> 1000 ->   10111 ->  a(8) = 23;
n =  9 -> 1001 ->  101110 ->  a(9) = 46;
n = 10 -> 1010 ->  101101 -> a(10) = 45;
n = 11 -> 1011 -> 1011010 -> a(11) = 90;
n = 12 -> 1100 ->  101011 -> a(12) = 43.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A179888, A005614.

import Data.List (unfoldr)
a196168 0 = 1
a196168 n = foldl (\v b -> (2 * v + 1)*(b + 1)) 0 $ reverse $ unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) n
   where r v b = (2 * v + 1)*(b+1)

Table[FromDigits[Flatten[IntegerDigits[n,2]/.{{0->1,1->{1,0}}}],2],{n,0,120}] (* Harvey P. Dale, Dec 12 2017 *)

def a(n):
    b = bin(n)[2:]
    return int(b.replace('1', 't').replace('0', '1').replace('t', '10'), 2)
print([a(n) for n in range(56)]) # Michael S. Branicky, Oct 28 2021

n = Sum_{i=0..1} b(i)*2^i with 0 <= b(i) <= 1, L >= 0, then a(n) = h(0,L) with h(v,i) = if i > L then v, otherwise h((2*v+1)*(b(i)+1),i-1).
From Jeffrey Shallit, Oct 28 2021: (Start)
a(n) satisfies the recurrences:
a(2n+1) = 2*a(2n)
a(4n) = -2*a(n) + 3*a(2n)
a(8n+2) = -8*a(n) + 8*a(2n) + a(4n+2)
a(8n+6) = -4*a(2n) + 5*a(4n+2)
which shows that a(n) is a 2-regular sequence. (End)

A099258 Inverse of A099257.

Original entry on oeis.org

1, 3, 2, 4, 6, 7, 8, 9, 5, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 11, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 23, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Reinhard Zumkeller, Oct 09 2004

nonn

Permutation of the natural numbers;
a(A033484(n)) = A033484(n);
a(A033484(n) - 1) = A033484(n)/2 = A055010(n).

Index entries for sequences that are permutations of the natural numbers

a(n) = if n = 3*2^k - 2 then n else (if n = 3*2^k - 3 then (n + 1)/2 else n + 1).

A266550 Independence number of the n-Mycielski graph.

Original entry on oeis.org

1, 1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 1

Eric W. Weisstein, Dec 31 2015

Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Mycielski Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052940, A055010, A083329, A153893.

[1,1] cat [-1+3*2^(n-3): n in [3..40]]; // Vincenzo Librandi, Jan 01 2016

I:=[1,1,2,5]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jan 01 2016

Table[Piecewise[{{-1 + 3 2^(n - 3), n > 2}}, 1], {n, 35}]
CoefficientList[Series[1 + x*(1 - x + x^2)/((1 - x)*(1 - 2*x)), {x, 0, 35}], x] (* Vincenzo Librandi, Jan 01 2016 *)

a(1) = 1, a(2) = 1; for n>2, a(n) = -1 + 3*2^(n-3) = A083329(n-2) = A055010(n-2) = A153893(n-3).
G.f.: x + x^2*(1 - x + x^2)/((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1)-2*a(n-2) for n>2. - Vincenzo Librandi, Jan 01 2016
a(n) = A052940(n-3) for n > 3. - Georg Fischer, Oct 23 2018
E.g.f.: (3*exp(2*x) - 8*exp(x) + 5 + 10*x+ 2*x^2)/8. - Stefano Spezia, Sep 14 2024

A332997 a(n) = A000120(A332995(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 05 2020

nonn

It seems that a(A055010(n)) = n for all n >= 0, and apart from n=1, A055010 seems to give the first occurrence of each n in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000120, A332817, A332897, A332995, A332998.

Cf. A052940, A055010.

A332997(n) = hammingweight(A332995(n));

PARI
```
A332997(n) = A332897(A332817(n));
```

a(n) = A000120(A332995(n)) = A332897(A332817(n)).

a(n) = A000120(n) - A332998(n).

A086652 a(n) = A000225(n+3)-A052955(n).

Original entry on oeis.org

6, 13, 28, 58, 120, 244, 496, 1000, 2016, 4048, 8128, 16288, 32640, 65344, 130816, 261760, 523776, 1047808, 2096128, 4192768, 8386560, 16774144, 33550336, 67102720, 134209536, 268423168, 536854528, 1073717248, 2147450880

Offset: 0

Marco Matosic, Jul 27 2003

Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A000225, A052955, A055010, A086221.

f:=n->2^(n+3)-((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4);

a(2n) = A006516(n+2); a(2n+1) = A086221(n+1).
G.f.: ( 6+x-10*x^2 ) / ( (2*x-1)*(2*x^2-1) ). - R. J. Mathar, Sep 15 2012
a(n) = 2^(n+3)-((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Sep 23 2014

Edited and extended by David Wasserman, Feb 17 2005

cp's OEIS Frontend

A086784 Number of non-trailing zeros in binary representation of n.

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A182461 a(n) = 3*a(n-1) - 2*a(n-2) with a(0)=16 and a(1)=40.

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A275726 A275725-polynomials evaluated at x=2: a(n) = A048675(A275725(n)).

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A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

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A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

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Mathematica

A196168 In binary representation of n: replace each 0 with 1, and each 1 with 10.

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A099258 Inverse of A099257.

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A266550 Independence number of the n-Mycielski graph.

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A182461 a(n) = 3a(n-1) - 2a(n-2) with a(0)=16 and a(1)=40.