A086893 -id:A086893 - cp's OEIS frontend

A061547 Number of 132 and 213-avoiding derangements of {1,2,...,n}.

1, 0, 1, 2, 6, 10, 26, 42, 106, 170, 426, 682, 1706, 2730, 6826, 10922, 27306, 43690, 109226, 174762, 436906, 699050, 1747626, 2796202, 6990506, 11184810, 27962026, 44739242, 111848106, 178956970, 447392426, 715827882, 1789569706, 2863311530, 7158278826

Offset: 0

Emeric Deutsch, May 16 2001

Or, number of permutations with no fixed points avoiding 213 and 132.
Number of derangements of {1,2,...,n} having ascending runs consisting of consecutive integers. Example: a(4)=6 because we have 234/1, 34/12, 34/2/1, 4/123, 4/3/12, 4/3/2/1, the ascending runs being as indicated. - Emeric Deutsch, Dec 08 2004
Let c be twice the sequence A002450 interlaced with itself (from the second term), i.e., c = 2*(0, 1, 1, 5, 5, 21, 21, 85, 85, 341, 341, ...). Let d be powers of 4 interlaced with the zero sequence: d = (1, 0, 4, 0, 16, 0, 64, 0, 256, 0, ...). Then a(n+1) = c(n) + d(n). - Creighton Dement, May 09 2005
Inverse binomial transform of A094705 (0, 1, 4, 15). - Paul Curtz, Jun 15 2008
Equals row sums of triangle A177993. - Gary W. Adamson, May 16 2010
a(n-1) is also the number of order preserving partial isometries (of an n-chain) of fix 1 (fix of alpha equals the number of fixed points of alpha). - Abdullahi Umar, Dec 28 2010
a(n+1) <= A218553(n) is also the Moore lower bound on the order of a (5,n)-cage. - Jason Kimberley, Oct 31 2011
For n > 0, a(n) is the location of the n-th new number to make a first appearance in A087230. E.g., the 17th number to make its first appearance in A087230 is 18 and this occurs at A087230(43690) and a(17)=43690. - K D Pegrume, Jan 26 2022
Position in A002487 of 2 adjacent terms of A000045. E.g., 3/5 at 10, 5/8 at 26, 8/13 at 42, ... - Ed Pegg Jr, Dec 27 2022

			a(4)=6 because the only 132 and 213-avoiding permutations of {1,2,3,4} without fixed points are: 2341, 3412, 3421, 4123, 4312 and 4321.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869 (contains the sequence of the odd-subscripted terms and that of the even-subscripted terms).
Emeric Deutsch, Derangements That Don't Rise Too Fast: 10902, Amer. Math. Monthly, Vol. 110, No. 7 (2003), pp. 639-640.
K. Dilcher and K. B. Stolarsky, Stern polynomials and double-limit continued fractions, Acta Arithmetica 140 (2009), 119-134
R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.0049 [math.GR], 2010.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Cf. A000166, A020988, A020989, A068018.
Cf. A177993. - Gary W. Adamson, May 16 2010
Cf. A183158, A183159. - Abdullahi Umar, Dec 28 2010
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), this sequence (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 31 2011

[(3/8)*2^n +(1/24)*(-2)^n - 2/3: n in [1..35]]; // Vincenzo Librandi, Aug 13 2011

A061547:=n->ceil(abs((3/8)*2^n +(1/24)*(-2)^n - 2/3)); seq(A061547(n), n=1..30); # Wesley Ivan Hurt, Apr 03 2014

f[n_] := (9*2^(n-3) - (-2)^(n-3) - 2)/3; Array[f, 32] (* Robert G. Wilson v, Aug 13 2011 *)

a(n)=(3/8)*2^n+(1/24)*(-2)^n-2/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3/8)*2^n + (1/24)*(-2)^n - 2/3 for n>=1.
a(n) = 4*a(n-2) + 2, a(0)=1, a(1)=0, a(2)=1.
G.f: (5*z^3-3*z^2-z+1)/((z-1)*(4*z^2-1)).
a(n) = A020989((n-2)/2) for n=2, 4, 6, ... and A020988((n-3)/2) for n=3, 5, 7, ... .
a(n+1)-2*a(n) = A078008 signed. Differences: doubled A000302. - Paul Curtz, Jun 15 2008
a(2i+1) = 2*Sum_{j=0..i-1} 4^j = string "2"^i read in base 4.
a(2i+2) = 4^i + 2*Sum_{j=0..i-1} 4^j = string "1"*"2"^i read in base 4.
a(n+2) = Sum_{k=0..n} A144464(n,k)^2 = Sum_{k=0..n} A152716(n,k). - Philippe Deléham and Michel Marcus, Feb 26 2014
a(2*n-1) = A176965(2*n), a(2*n) = A176965(2*n-1) for n>0. - Yosu Yurramendi, Dec 23 2016
a(2*n-1) = A020988(k-1), a(2*n)= A020989(n-1) for n>0. - Yosu Yurramendi, Jan 03 2017
a(n+2) = 2*A086893(n), n > 0. - Yosu Yurramendi, Mar 07 2017
E.g.f.: (15 - 8*cosh(x) + 5*cosh(2*x) - 8*sinh(x) + 4*sinh(2*x))/12. - Stefano Spezia, Apr 07 2022

a(0)=1 prepended by Alois P. Heinz, Jan 27 2022

A371094 a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.

Original entry on oeis.org

1, 21, 7, 21, 13, 341, 19, 45, 25, 117, 31, 69, 37, 341, 43, 93, 49, 213, 55, 117, 61, 5461, 67, 141, 73, 309, 79, 165, 85, 725, 91, 189, 97, 405, 103, 213, 109, 1877, 115, 237, 121, 501, 127, 261, 133, 1109, 139, 285, 145, 597, 151, 309, 157, 5461, 163, 333, 169, 693, 175, 357, 181, 1493, 187, 381, 193, 789, 199

Offset: 0

Antti Karttunen (proposed by Ali Sada), Apr 19 2024

Construction: take the binary expansion of 3n+1 (A016777(n)), and substitute "01" for all trailing 0-bits that follow after its odd part (= A067745(1+n)), of which there are A371093(n) in total. See the examples.

			For n=1, 3*n+1 = 4, "100" in binary, when we substitute 01's for the two trailing 0's, we obtain 21, "10101" in binary, therefore a(1) = 21.
For n=6, 3*6+1 = 19, "10011" in binary, and there are no trailing 0's, and no changes, therefore a(6) = 19.
For n=7, 3*7+1 = 22, "10110" in binary, with one trailing 0, which when replaced with 01 gives us 45, "101101" in binary, therefore a(7) = 45.
For n=229, there are e=4 trailing bit expansions 0 -> 01,
  3n+1 = binary  101011  0 0 0 0
  a(n) = binary  101011 01010101

Antti Karttunen, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Plot binary expansion of a(n) arranged with smallest bit at bottom and largest at top, n increasing from left to right for n = 0..3071. Black indicates 1, white indicates 0.
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences related to binary expansion of n

Cf. A016777, A067745, A075677, A086893, A096773, A372289, A371093.
Cf. A016921, A372351 (even and odd bisection), A372290 (numbers occurring in the latter).
Cf. arrays A372282, A372283 and A371096, A371100, A371102.
Cf. also A302338.

Array[#2*(2^#3) + ((4^#3) - 1)/3 & @@ {#1, #2, IntegerExponent[#2, 2]} & @@ {#, 3 #1 + 1} &, 67, 0] (* Michael De Vlieger, Apr 19 2024 *)

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };

def A371094(n): return ((m:=3*n+1)<<(e:=(~m & m-1).bit_length()))+((1<<(e<<1))-1)//3 # Chai Wah Wu, Apr 28 2024

a(n) = A372289(A016777(n)).

a(2n) = A016777(2n) = A016921(n).

A039004 Numbers whose base-4 representation has the same number of 1's and 2's.

Original entry on oeis.org

0, 3, 6, 9, 12, 15, 18, 24, 27, 30, 33, 36, 39, 45, 48, 51, 54, 57, 60, 63, 66, 72, 75, 78, 90, 96, 99, 102, 105, 108, 111, 114, 120, 123, 126, 129, 132, 135, 141, 144, 147, 150, 153, 156, 159, 165, 177, 180, 183, 189, 192, 195, 198, 201, 204, 207, 210, 216, 219

Offset: 1

Olivier Gérard

Numbers such that sum (-1)^k*b(k) = 0 where b(k)=k-th binary digit of n (see A065359). - Benoit Cloitre, Nov 18 2003
Conjecture: a(C(2n,n)-1) = 4^n - 1. (A000984 is C(2n,n)). - Gerald McGarvey, Nov 18 2007
From Russell Jay Hendel, Jun 23 2015: (Start)
We prove the McGarvey conjecture (A) a(e(n,n)-1) = 4^n-1, with e(n,m) = A034870(n,m) = binomial(2n,m), the even rows of Pascal's triangle. By the comment from Hendel in A034870, we have the function s(n,k) = #{n-digit, base-4 numbers with n-k more 1-digits than 2-digits}. As shown in A034870, (B) #s(n,k)= e(n,k) with # indicating cardinality, that is, e(n,k) = binomial(2n,k) gives the number of n-digit, base-4 numbers with n-k more 1-digits than 2-digits.
We now show that (B) implies (A). By definition, s(n,n) contains the e(n,n) = binomial(2n,n) numbers with an equal number of 1-digits and 2-digits. The biggest n-digit, base-4 number is 333...3 (n copies of 3). Since 333...33 has zero 1-digits and zero 2-digits it follows that 333...333 is a member of s(n,n) and hence it is the biggest member of s(n,n). But 333...333 (n copies of 3) in base 4 has value 4^n-1. Since A039004 starts with index 0 (that is, 0 is the 0th member of A039004), it immediately follows that 4^n-1 is the (e(n,n)-1)st member of A039004, proving the McGarvey conjecture. (End)
Also numbers whose alternating sum of binary expansion is 0, i.e., positions of zeros in A345927. These are numbers whose binary expansion has the same number of 1's at even positions as at odd positions. - Gus Wiseman, Jul 28 2021

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A139370, A139371, A139372, A139373.
Cf. A139351, A139352, A139353, A139354, A139355.
A subset of A001969 (evil numbers).
A base-2 version is A031443 (digitally balanced numbers).
Positions of 0's in A065359 and A345927.
Positions of first appearances are A086893.
The version for standard compositions is A344619.
A000120 and A080791 count binary digits, with difference A145037.
A003714 lists numbers with no successive binary indices.
A011782 counts compositions.
A030190 gives the binary expansion of each nonnegative integer.
A070939 gives the length of an integer's binary expansion.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A101211 lists run-lengths in binary expansion:
- row-lengths: A069010
- reverse: A227736
- ones only: A245563
A138364 counts compositions with alternating sum 0:
- bisection: A001700/A088218
- complement: A058622
A328594 lists numbers whose binary expansion is aperiodic.
A345197 counts compositions by length and alternating sum.
Cf. A000302, A000346, A003242, A029837, A053738, A053754, A071321, A103919, A191232, A328596.

Fortran
```
c See link in A139351.
```

N:= 1000: # to get all terms up to N, which should be divisible by 4
B:= Array(0..N-1):
d:= ceil(log[4](N));
S:= Array(0..N-1,[seq(op([0,1,-1,0]),i=1..N/4)]):
for i from 1 to d do
  B:= B + S;
  S:= Array(0..N-1,i-> S[floor(i/4)]);
od:
select(t -> B[t]=0, [$0..N-1]); # Robert Israel, Jun 24 2015

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[IntegerDigits[#,2]]==0&] (* Gus Wiseman, Jul 28 2021 *)

for(n=0,219,if(sum(i=1,length(binary(n)),(-1)^i*component(binary(n),i))==0,print1(n,",")))

Conjecture: there is a constant c around 5 such that a(n) is asymptotic to c*n. - Benoit Cloitre, Nov 24 2002

That conjecture is false. The number of members of the sequence from 0 to 4^d-1 is binomial(2d,d) which by Stirling's formula is asymptotic to 4^d/sqrt(Pi*d). If Cloitre's conjecture were true we would have 4^d-1 asymptotic to c*4^d/sqrt(Pi*d), a contradiction. - Robert Israel, Jun 24 2015

A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

Original entry on oeis.org

1, 21, 3, 5461, 21, 5, 357913941, 5461, 341, 7, 1537228672809129301, 357913941, 1398101, 45, 9, 28356863910078205288614550619314017621, 1537228672809129301, 23456248059221, 1109, 117, 11, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 28356863910078205288614550619314017621, 6602346876188694799461995861, 873813, 11605, 69, 13

Offset: 1

Antti Karttunen, Apr 28 2024

			Array begins:
n\k|    1     2        3     4      5     6        7     8      9     10
---+----------------------------------------------------------------------
1  |    1,    3,       5,    7,     9,   11,      13,   15,    17,    19,
2  |   21,   21,     341,   45,   117,   69,     341,   93,   213,   117,
3  | 5461, 5461, 1398101, 1109, 11605, 3413, 1398101, 2261, 87381, 11605,

Cf. A005408 (row 1), A372351 (row 2, bisection of A371094), A372444 (column 14).
Arrays derived from this one:
  A372283,
  A372285 the number of terms of A086893 in the interval [A(n, k), A(1+n, k)],
  A372287 the column index of A(n, k) in array A257852,
  A372288 the sum of digits of A(n, k) in "Jacobsthal greedy base",
  A372353 differences between A(n,k) and the largest term of A086893 <= A(n,k),
  A372354 floor(log_2(.)) of terms, A372356 (and their columnwise first differences),
  A372359 terms xored with binary words of the same length, either of the form 10101...0101 or 110101...0101, depending on whether the binary length is odd or even.
Cf. also arrays A371096, A371102 that give subsets of columns of this array, and array A371100 that gives the terms of the row 2 in different order.

up_to = 28;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372282sq(n,k) = if(1==n,2*k-1,A371094(A372282sq(n-1,k)));
A372282list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372282sq((a-(col-1)),col))); (v); };
v372282 = A372282list(up_to);
A372282(n) = v372282[n];

A096773 a(n) = 4*a(n-2) + 1 with a(1) = 0, a(2) = 3.

Original entry on oeis.org

0, 3, 1, 13, 5, 53, 21, 213, 85, 853, 341, 3413, 1365, 13653, 5461, 54613, 21845, 218453, 87381, 873813, 349525, 3495253, 1398101, 13981013, 5592405, 55924053, 22369621, 223696213, 89478485, 894784853, 357913941, 3579139413, 1431655765

Offset: 1

Gottfried Helms, Aug 15 2004

Remainders for classes m of integers n (mod 2^(m+1)). After applying one Collatz (3x+1)-transformation to the so-classified integers the result can be written in two classes (mod 6) only.
This classifying scheme covers all positive integers.
With one 3x+1-transformation T(x;p) := x' = (3x+1)/2^p, all numbers x, described in the form, with the free parameter i >= 0, x = i*2^N + a(N) result in x', describable by the two classes with the same parameter i:
x' = i*6 + 1 (for odd N>2), or x' = i*6 + 5 (for even N). Thus
x =  4*i +  3 -> x' = 6*i + 5, x =   8*i +  1 -> x' = 6*i + 1,
x = 16*i + 13 -> x' = 6*i + 5, x =  32*i +  5 -> x' = 6*i + 1,
x = 64*i + 53 -> x' = 6*i + 5, x = 128*i + 21 -> x' = 6*i + 1,
....
all with "i" as a free parameter >= 0 covering all positive integers.

			a(1) = (2^0-1)/3 =  0, a(2) = (5*2^1 - 1) / 3 =  3,
a(3) = (2^2-1)/3 =  1, a(4) = (5*2^3 - 1) / 3 = 13,
a(5) = (2^4-1)/3 =  5, a(6) = (5*2^5 - 1) / 3 = 53,
a(7) = (2^6-1)/3 = 21.
....

Michael De Vlieger, Table of n, a(n) for n = 1..3323
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Bisections are A002450 & A072197.
After the initial 0, column 1 of A257852.
Cf. A176965.

[(2^(n-1)*(3 + 2*(-1)^n) - 1)/3: n in [1..40]]; // Vincenzo Librandi, Jul 12 2015

a[1] = 0; a[2] = 3; a[n_] := a[n] = 4a[n - 2] + 1; Table[ a[n], {n, 35}] (* Robert G. Wilson v, Aug 20 2004 *)
Table[(2^(n - 1)*(3 + 2*(-1)^(n)) - 1)/3, {n, 10}] (* L. Edson Jeffery, Jul 12 2015 *)
nxt[{a_,b_}]:={b,4a+1}; NestList[nxt,{0,3},40][[;;,1]] (* or *) LinearRecurrence[{1,4,-4},{0,3,1},40] (* Harvey P. Dale, Mar 19 2025 *)

apply( {A096773(n) = if(n%2, 1, 5)<<(n-1)\3}, [1..55]) \\ M. F. Hasler, May 28 2024

# To map any (odd) v to its (r,c):
use bigint; $v=149; $r=$c=0; while(1){ $b=($v&1); $v>>=1; if ($b==($v&1)){ $c=($v>>1); last} $r++} $r&=1; # this splits the binary representation into two parts, at the first repeated digit from the right: the number of bits on the right is the row value, and the binary value on the left is the column value. Example: 149 => 1.00.10101 => (r,c)=(5,1). Ruud H.G. van Tol, Sep 23 2021

A096773=lambda n:((1 if n&1 else 5)<M. F. Hasler, May 28 2024

a(2m) = (5*2^(2m-1) - 1)/3, a(2m-1) = (2^(2m-2)-1)/3.
From Paul Curtz, Jul 01 2008; corrected by Bob Selcoe, Jul 28 2018: (Start)
a(2n) = 10*a(2n-1) + 3.
a(n+1) - 2*a(n) = A001045(n+2), signed. (End)
a(n) = (2^(n-1)*(3 + 2*(-1)^n) - 1)/3. - L. Edson Jeffery, Jul 12 2015
a(2n) = A086893(2n), a(2n+1) = A086893(2n-1), n > 0. - Yosu Yurramendi, Jan 17 2017
G.f.: -x^2*(-3+2*x) / ( (x-1)*(2*x+1)*(2*x-1) ). - R. J. Mathar, Mar 07 2017
a(2n) = A072197(n-1), n > 0; a(2n+1) = A002450(n), n >= 0. - Yosu Yurramendi, Mar 07 2017
a(2n) = (A266753(n) + A004171(n-1))/2, a(2n+1) = (A266753(n) - A004171(n-1))/2, n > 0. - Yosu Yurramendi, Mar 07 2017
a(n) = least residue 2*3^(2^(n-4)-1) - 1 (mod 2^n), n >= 5. - Bob Selcoe, Jul 26 2018
a(n) = 2*A176965(n-1) + 1 for n > 1. - Loren M. Pearson, Dec 06 2024

A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

Original entry on oeis.org

27, 41, 31, 47, 71, 107, 161, 121, 91, 137, 103, 155, 233, 175, 263, 395, 593, 445, 167, 251, 377, 283, 425, 319, 479, 719, 1079, 1619, 2429, 911, 1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 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

Offset: 0

Antti Karttunen, May 01 2024

nonn

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000265, A075677, A371093.
Column 14 of A372283, Row 13 of A256598 (but only up to the first 1).
Row 1 of A372560.
From term 47 to the first 1 same as A088593.
Sequences derived from this one or related to:
  A372444,
  A372445 column index of a(n) in array A257852,
  A372362 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372447 binary lengths minus 1,
  A372446 a(n) xored with the term of A086893 having the same binary length,
  A372453 a(n) minus the term of A086893 having the same binary length.

R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372443(n) = { my(x=27); while(n, x=R(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = R(a(n-1)), where R(n) = (3*n+1)/2^A371093(n) = A000265(3*n+1).

For n > 0, a(n) = R(A372444(n-1)) = A000265(1+3*A372444(n-1)).

A319420 Irregular triangle read by rows: row n lists the cuts-resistances of the 2^n binary vectors of length n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 22 2018

The cuts-resistance of a vector is defined in A319416. The 2^n vectors of length n are taken in lexicographic order.
Note that here the vectors can begin with either 0 or 1, whereas in A319416 only vectors beginning with 1 are considered (since there we are considering binary representations of numbers).
Conjecture: The row sums, halved, appear to match A189391.

			Triangle begins:
0,
1,1,
2,1,1,2,
3,2,1,2,2,1,2,3,
4,3,2,2,2,1,2,3,3,2,1,2,2,2,3,4,
5,4,3,3,3,2,2,3,3,2,1,2,2,2,3,4,4,3,2,2,2,1,2,3,3,2,2,2,3,3,3,4,5,
...

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See table on page 4.

Keeping the first digit gives A319416.
Positions of 1's are the terms > 1 of A061547 and A086893, all minus 1.
The version for runs-resistance is A329870.
Compositions counted by cuts-resistance are A329861.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A027383, A189391, A318921, A318928, A319411, A329767, A329862, A329865.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[degdep[Rest[IntegerDigits[n,2]]],{n,0,50}] (* Gus Wiseman, Nov 25 2019 *)

A372444 The n-th iterate of 27 with A371094.

Original entry on oeis.org

27, 165, 8021, 12408149, 19607957362005, 32439509492992549521282389, 58947232705679751034215288252890081792789279233365, 259166427025070423330595967015238989905128148712607202753574381749095993394717720069452733214971221

Offset: 0

Antti Karttunen, May 01 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10

Cf. A371094.
Column 7 of A371102, column 14 of A372282.
Column 1 of A372560.
Sequences derived from this one:
  A372443 obtained when Reduced Collatz-function R is applied to a(n-1), for n > 0,
  A372445 column index of a(n) in array A257852,
  A372448 the 2-adic valuation of 1 + 3*a(n), equal to row index of a(n) in array A257852,
  A372449 binary lengths minus 1; their first differences: A372451,
  A372452 number of terms of A086893 in the interval [a(n), a(1+n)],
  A372454 the difference between a(n) and the term of A086893 with the same binary length.

A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A372444(n) = { my(x=27); while(n, x=A371094(x); n--); (x); };

a(0) = 27; for n > 0, a(n) = A371094(a(n-1)).

A283641 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 678", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 11, 101, 1101, 10101, 110101, 1010101, 11010101, 101010101, 1101010101, 10101010101, 110101010101, 1010101010101, 11010101010101, 101010101010101, 1101010101010101, 10101010101010101, 110101010101010101, 1010101010101010101, 11010101010101010101

Offset: 0

Robert Price, Mar 12 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
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 sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A266508, A086893, A283642.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 678; 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[1, i]], 10], {i, 1, stages - 1}]

Conjectures from Colin Barker, Mar 14 2017: (Start)
G.f.: (1 + 10*x - 10*x^2) / ((1 - x)*(1 - 10*x)*(1 + 10*x)).
a(n) = (-2 - 9*(-10)^n + 209*10^n) / 198.
a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n>2.
(End)

A345927 Alternating sum of the binary expansion of n (row n of A030190). Replace 2^k with (-1)^(A070939(n)-k) in the binary expansion of n (compare to the definition of A065359).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 14 2021

sign

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The binary expansion of 53 is (1,1,0,1,0,1), so a(53) = 1 - 1 + 0 - 1 + 0 - 1 = -2.

Binary expansions of each nonnegative integer are the rows of A030190.
The positions of 0's are A039004.
The version for prime factors is A071321 (reverse: A071322).
Positions of first appearances are A086893.
The version for standard compositions is A124754 (reverse: A344618).
The version for prime multiplicities is A316523.
The version for prime indices is A316524 (reverse: A344616).
A003714 lists numbers with no successive binary indices.
A070939 gives the length of an integer's binary expansion.
A103919 counts partitions by sum and alternating sum.
A328594 lists numbers whose binary expansion is aperiodic.
A328595 lists numbers whose reversed binary expansion is a necklace.
Cf. A000037, A000070, A000120, A027187, A028260, A065359, A069010, A116406, A121016, A191232, A245563, A344609, A344619.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[ats[IntegerDigits[n,2]],{n,0,100}]

a(n) = subst(Pol(Vecrev(binary(n))), x, -1); \\ Michel Marcus, Jul 19 2021

def a(n): return sum((-1)**k for k, bi in enumerate(bin(n)[2:]) if bi=='1')
print([a(n) for n in range(84)]) # Michael S. Branicky, Jul 19 2021

a(n) = (-1)^(A070939(n)-1)*A065359(n).

cp's OEIS Frontend

A061547 Number of 132 and 213-avoiding derangements of {1,2,...,n}.

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A371094 a(n) = m*(2^e) + ((4^e)-1)/3, where m = 3n+1, and e is the 2-adic valuation of m.

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A039004 Numbers whose base-4 representation has the same number of 1's and 2's.

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A372282 Array read by upward antidiagonals: A(n, k) = A371094(A(n-1, k)) for n > 1, k >= 1; A(1, k) = 2*k-1.

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A096773 a(n) = 4*a(n-2) + 1 with a(1) = 0, a(2) = 3.

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A372443 The n-th iterate of 27 with Reduced Collatz-function R, which gives the odd part of 3n+1.

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A319420 Irregular triangle read by rows: row n lists the cuts-resistances of the 2^n binary vectors of length n.

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