A081253 -id:A081253 - cp's OEIS frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

1, 3, 6, 13, 26, 53, 106, 213, 426, 853, 1706, 3413, 6826, 13653, 27306, 54613, 109226, 218453, 436906, 873813, 1747626, 3495253, 6990506, 13981013, 27962026, 55924053, 111848106, 223696213, 447392426, 894784853, 1789569706, 3579139413

Offset: 1

Klaus Brockhaus, Mar 17 2003

The limit of the local maxima, lim_{m->inf} A081252(m)/m^2 = 1/10. For local minima cf. A081253.

Row sums of the triangle A181971. - Reinhard Zumkeller, Jul 09 2012

			13 is a term since A081252(12)/12^2 = 15/144 = 0.104..., A081252(13)/13^2 = 18/169 = 0.106..., A081252(14)/14^2 = 20/196 = 0.102....

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A020714, A020989, A048573, A052549, A053646, A072197, A081252, A081253, A266219 (binary).

[Floor(2^(n-1)*5/3): n in [1..40]]; // Vincenzo Librandi, Apr 04 2012

seq(floor(2^(n-1)*5/3),n=1..35); # Muniru A Asiru, Sep 20 2018

Rest@CoefficientList[Series[-(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)), {x, 0, 32}], x] (* Vincenzo Librandi, Apr 04 2012 *)
a[n_]:=Floor[2^(n-1)*5/3]; Array[a,33,1] (* Stefano Spezia, Sep 01 2018 *)

a(n) = 2^(n-1)*5\3; \\ Altug Alkan, Sep 21 2018

a(n) = floor(2^(n-1)*5/3). [corrected by Michel Marcus, Sep 21 2018]
a(n) = a(n-2) + 5*2^(n-3) for n > 2;
a(n+2) - a(n) = A020714(n-1);
a(n) + a(n-1) = A052549(n-1) for n > 1;
a(2*n+1) = A020989(n); a(2n) = A072197(n-1);
a(n+1) - a(n) = A048573(n-1).
G.f.: -(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)).
a(n) = 5*2^(n-1)/3 + (-1)^n/6-1/2. a(n) = 2*a(n-1) + (1+(-1)^n)/2, a(1)=1. - Paul Barry, Mar 24 2003
a(2n) = 2*a(2*n-1) + 1, a(2*n+1) = 2*a(2*n), a(1)=1. a(n) = A000975(n-1) + 2^(n-1). - Philippe Deléham, Oct 15 2006
a(n) = A005578(n) + A000225(n-1). - Yuchun Ji, Sep 21 2018
a(n) - a(n-2) = 2 * (a(n-1) - a(n-3)), with a(0..2)=[1,3,6]. - Yuchun Ji, Mar 18 2020

A053646 Distance to nearest power of 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Offset: 1

Henry Bottomley, Mar 22 2000

Sum_{j=1..2^(k+1)} a(j) = A002450(k) = (4^k - 1)/3. - Klaus Brockhaus, Mar 17 2003

			a(10)=2 since 8 is closest power of 2 to 10 and |8-10| = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Hsien-Kuei Hwang, S. Janson, and T.-H. 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.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, 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.
Index entries for sequences related to distance to nearest element of some set

Cf. A006165, A053188, A060973, A081134, A002450, A081252, A081253, A081254.

a:= n-> (h-> min(n-h, 2*h-n))(2^ilog2(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2021

np2[n_]:=Module[{min=Floor[Log[2,n]],max},max=min+1;If[2^max-nHarvey P. Dale, Feb 21 2012 *)

a(n)=vecmin(vector(n,i,abs(n-2^(i-1))))

for(n=1,89,p=2^floor(0.1^25+log(n)/log(2)); print1(min(n-p,2*p-n),","))

a(n) = my (p=#binary(n)); return (min(n-2^(p-1), 2^p-n)) \\ Rémy Sigrist, Mar 24 2018

def A053646(n): return min(n-(m := 2**(len(bin(n))-3)),2*m-n) # Chai Wah Wu, Mar 08 2022

a(2^k+i) = i for 1 <= i <= 2^(k-1); a(3*2^k+i) = 2^k-i for 1 <= i <= 2^k; (Sum_{k=1..n} a(k))/n^2 is bounded. - Benoit Cloitre, Aug 17 2002
a(n) = min(n-2^floor(log(n)/log(2)), 2*2^floor(log(n)/log(2))-n). - Klaus Brockhaus, Mar 08 2003
From Peter Bala, Aug 04 2022: (Start)
a(n) = a( 1 + floor((n-1)/2) ) + a( ceiling((n-1)/2) ).
a(2*n) = 2*a(n); a(2*n+1) = a(n) + a(n+1) for n >= 2. Cf. A006165. (End)
a(n) = 2*A006165(n) - n for n >= 2. - Peter Bala, Sep 25 2022

A206374 a(n) = (7*4^n - 1)/3.

Original entry on oeis.org

2, 9, 37, 149, 597, 2389, 9557, 38229, 152917, 611669, 2446677, 9786709, 39146837, 156587349, 626349397, 2505397589, 10021590357, 40086361429, 160345445717, 641381782869, 2565527131477, 10262108525909, 41048434103637, 164193736414549, 656774945658197

Offset: 0

Brad Clardy, Feb 07 2012

First bisection of A062092 and A081253, second bisection of A097163. - Bruno Berselli, Feb 12 2012

Except a(0)=2, this is the 3rd row of table A178415. - Michel Marcus, Apr 13 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mike Warburton, Ulam-Warburton Automaton - Counting Cells with Quadratics, arXiv:1901.10565 [math.CO], 2019. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002450, A006666, A072197; A002042 (first differences), A178415, A347834.

Magma
```
[(7*4^n-1)/3 : n in [0..30]];
```

Table[(7(4^n) - 1)/3, {n, 0, 24}] (* Alonso del Arte, Feb 11 2012 *)
CoefficientList[Series[(2-x)/(1-5*x+4*x^2),{x,0,30}],x] (* or *) LinearRecurrence[{5,-4},{2,9},30] (* Vincenzo Librandi, Mar 20 2012 *)

vector(20,n,(7*4^(n-1)-1)/3) \\ Derek Orr, Apr 12 2015

G.f.: (2-x)/(1-5*x+4*x^2). - Bruno Berselli, Feb 12 2012
a(n) = A083597(n)+1. - Bruno Berselli, Feb 12 2012
a(n) = 4*a(n-1)+1 for n>0, a(0)=2. - Bruno Berselli, Oct 22 2015
a(n) = 7*A002450(n) + 2. - Yosu Yurramendi, Jan 24 2017
A006666(a(n)) = 2*n+11 for n > 0. - Juan Miguel Barga Pérez, Jun 18 2020
a(n) = 5*a(n-1) - 4*a(n-2) for n >= 2. - Wesley Ivan Hurt, Jun 30 2020
a(n) = A178415(3, n) = A347834(4, n-1), arrays, for n >= 1.- Wolfdieter Lang, Nov 29 2021

A081252 Partial sums of A053646.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 5, 6, 8, 11, 15, 18, 20, 21, 21, 22, 24, 27, 31, 36, 42, 49, 57, 64, 70, 75, 79, 82, 84, 85, 85, 86, 88, 91, 95, 100, 106, 113, 121, 130, 140, 151, 163, 176, 190, 205, 221, 236, 250, 263, 275, 286, 296, 305, 313, 320, 326, 331, 335, 338, 340, 341, 341

Offset: 1

Klaus Brockhaus, Mar 17 2003

			First seven terms of A053646 are 0,0,1,0,1,2,1, so a(7) = 5.

Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254

Cf. A053646, A081253, A081254.

{s=0; for(n=1,65,p=2^floor(0.1^25+log(n)/log(2)); print1(s=s+min(n-p,2*p-n),","))}

a(n) = sum{j=1..n, A053646(j)}.

A266071 Binary representation of the middle column of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 10, 100, 1001, 10010, 100101, 1001010, 10010101, 100101010, 1001010101, 10010101010, 100101010101, 1001010101010, 10010101010101, 100101010101010, 1001010101010101, 10010101010101010, 100101010101010101, 1001010101010101010, 10010101010101010101

Offset: 0

Robert Price, Dec 20 2015

			From _Michael De Vlieger_, Dec 21 2015: (Start)
First 8 rows at left with the center column values in parentheses, and at right the binary value of center column cells up to that row:
              (1)               ->         1
            1 (0) 0             ->        10
          0 0 (0) 1 0           ->       100
        1 1 1 (1) 0 0 1         ->      1001
      0 0 0 0 (0) 0 1 0 0       ->     10010
    1 1 1 1 1 (1) 1 0 0 1 1     ->    100101
  0 0 0 0 0 0 (0) 0 0 1 0 0 0   ->   1001010
1 1 1 1 1 1 1 (1) 1 1 0 0 1 1 1 ->  10010101
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (10,1,-10).

Cf. A266070, A081253 (decimal).

Table[SeriesCoefficient[(1 - x^2 + x^3)/(1 - 10 x - x^2 + 10 x^3), {x, 0, n}], {n, 0, 19}] (* Michael De Vlieger, Dec 21 2015 *)

print([991*10**n//990 for n in range(50)]) # Karl V. Keller, Jr., Oct 09 2021

G.f.: (1 - x^2 + x^3)/(1 - 10*x - x^2 + 10*x^3). - Michael De Vlieger, Dec 21 2015

a(n) = floor(991*10^n/990). - Karl V. Keller, Jr., Oct 09 2021

A266613 Decimal representation of the middle column of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 5, 10, 20, 41, 82, 165, 330, 661, 1322, 2645, 5290, 10581, 21162, 42325, 84650, 169301, 338602, 677205, 1354410, 2708821, 5417642, 10835285, 21670570, 43341141, 86682282, 173364565, 346729130, 693458261, 1386916522, 2773833045, 5547666090, 11095332181

Offset: 0

Robert Price, Jan 01 2016

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000975, A266608, A266611, A266612, A081253.

# Rule 41: value in generation r and column c, where c=0 is the central one
r41 := proc(r::integer,c::integer)
    option remember;
    local up ;
    if r = 0 then
        if c = 0 then
            1;
        else
            0;
        end if;
    else
        # previous 3 bits
        [procname(r-1,c+1),procname(r-1,c),procname(r-1,c-1)] ;
        up := op(3,%)+2*op(2,%)+4*op(1,%) ;
        # rule 41 = 00101001_2: {5,3,0}->1, all others ->0
        if up in {5,3,0} then
            1;
        else
            0 ;
        end if;
    end if;
end proc:
A266613 := proc(n)
    b := [seq(r41(r,0),r=0..n)] ;
    add(op(-i,b)*2^(i-1),i=1..nops(b)) ;
end proc:
smax := 20 ;
L := [seq(A266613(n),n=0..smax)] ; # R. J. Mathar, Apr 12 2019

rule=41; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k],2],{k,1,rows}] (* Binary Representation of Middle Column *)

A266612(n) = A007088(a(n)).
Conjectures from Colin Barker, Jan 02 2016 and Apr 16 2019: (Start)
a(n) = (31*2^n-4*((-1)^n+3))/24 for n>2.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3) for n>5. - [corrected by Karl V. Keller, Jr., Oct 07 2021]
G.f.: (1-x^4+x^5) / ((1-x)*(1+x)*(1-2*x)). (End)
Conjecture: a(n) = A000975(n) + 20*2^(n-5), for n>2. - Andres Cicuttin, Mar 31 2016

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - N. J. A. Sloane, Jun 13 2022

A191665 Dispersion of A042963 (numbers >1, congruent to 1 or 2 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 6, 4, 10, 13, 9, 7, 21, 26, 18, 14, 8, 42, 53, 37, 29, 17, 11, 85, 106, 74, 58, 34, 22, 12, 170, 213, 149, 117, 69, 45, 25, 15, 341, 426, 298, 234, 138, 90, 50, 30, 16, 682, 853, 597, 469, 277, 181, 101, 61, 33, 19, 1365, 1706, 1194, 938, 554

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000975
Row 2:  A081254
Row 3:  A081253
Row 4:  A052997
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.

			Northwest corner:
1...2...5....10...21
3...6...13...26...53
4...9...18...37...74
7...14..29...58...117
8...17..34...69...138

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

Cf. A042963, A014601, A191426, A191664.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 2; b = 5; 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}]  (* A042963: (2+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}]]
(* A191665 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191665  *)

A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 2, 121, 4, 2035, 8, 32743, 16, 524239, 32, 8388511, 64, 134217535, 128, 2147483263, 256, 34359737599, 512, 549755812351, 1024, 8796093019135, 2048, 140737488349183, 4096, 2251799813672959, 8192, 36028797018939391, 16384, 576460752303374335, 32768

Offset: 0

Robert Price, Dec 20 2015

Rule 35 also generates this sequence.

			From _Michael De Vlieger_, Dec 21 2015: (Start)
First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:
              1                =               1 ->     1
            1 0 0              =             100 ->     4
          . . . 1 0            =              10 ->     2
        1 1 1 1 0 0 1          =         1111001 ->   121
      . . . . . . 1 0 0        =             100 ->     4
    1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035
  . . . . . . . . . 1 0 0 0    =            1000 ->     8
1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074.

rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]

print([2*4**n - 3*2**((n-1)//2) - 1 if n%2 else 2**(n//2) for n in range(30)]) # Karl V. Keller, Jr., Aug 25 2021

G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 and Apr 18 2019

a(n) = 2*4^n - 3*2^((n-1)/2) - 1 for odd n; a(n) = 2^(n/2) for even n. - Karl V. Keller, Jr., Aug 25 2021

A266255 Decimal representation of the n-th iteration of the "Rule 11" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 3, 124, 3, 2044, 3, 32764, 3, 524284, 3, 8388604, 3, 134217724, 3, 2147483644, 3, 34359738364, 3, 549755813884, 3, 8796093022204, 3, 140737488355324, 3, 2251799813685244, 3, 36028797018963964, 3, 576460752303423484, 3, 9223372036854775804, 3

Offset: 0

Robert Price, Dec 25 2015

Rule 43 also generates this sequence.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A266253, A266254, A266070, A266071, A081253, A266256, A266257, A266258, A266259.

rule=11; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

print([2*4**n - 4 if n%2 else 3 - 2*0**n for n in range(33)]) # Karl V. Keller, Jr., Aug 26 2021

From Colin Barker, Dec 27 2015 and Apr 14 2019: (Start)
a(n) = (7*(-1)^n+2^(2*n+1)-(-1)^n*2^(2*n+1)-1)/2 for n>0.
a(n) = 17*a(n-2)-16*a(n-4) for n>4.
G.f.: (1+4*x-14*x^2+56*x^3-32*x^4) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)).
(End)
a(n) = 2*4^n - 4 for odd n; a(n) = 3 - 2*0^n for even n. - Karl V. Keller, Jr., Aug 26 2021

A294523 Lexicographically earliest sequence of positive terms, such that, for any n > 0, the binary expansion of n, say of size k+1, is (1, a(n) mod 2, a^2(n) mod 2, ..., a^k(n) mod 2) (where a^i denotes the i-th iterate of the sequence).

Original entry on oeis.org

1, 2, 1, 2, 6, 5, 1, 2, 10, 6, 14, 9, 5, 13, 1, 2, 18, 10, 22, 12, 6, 14, 30, 17, 9, 5, 11, 25, 13, 29, 1, 2, 34, 18, 38, 20, 10, 22, 46, 24, 12, 6, 54, 28, 14, 30, 62, 33, 17, 9, 19, 41, 5, 11, 23, 49, 25, 13, 27, 57, 29, 61, 1, 2, 66, 34, 70, 36, 18, 38, 78

Offset: 1

Rémy Sigrist, Nov 01 2017

More informally, the parity of the iterate of the sequence at n gives the binary expansion of n (beyond the leading 1).
Apparently, iterating the sequence always leads to one of these three loops:
- the fixed point (1) iff we start from 2^k-1 for some k > 0,
- the fixed point (2) iff we start from 2^k for some k > 0,
- or (5, 6) for any other starting value.
a(n) is even iff n belongs to A004754.
a(n) is odd iff n belongs to A004760.
If a(n) > n then a(n) = A080541(n).
If n < 2^k then a(n) < 2^k.
Apparently, if a(n) > 2, then A054429(a(n)) = a(A054429(n)); this accounts for the symmetry of the part connected to the loop (5,6) in the oriented graph of this sequence.

			For n=11:
- the binary representation of 11 is (1,0,1,1),
- a(11) = 14 has parity 0,
- a(14) = 13 has parity 1,
- a(13) = 5 has parity 1,
- we find the binary digits of 11 beyond the initial 1, in order: 0, 1, 1.
See also representations of first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, Representation of the first 32 terms as an oriented graph (n -> a(n))
Rémy Sigrist, Representation of the first 64 terms as an oriented graph (n -> a(n))
Rémy Sigrist, PARI program for A294523

Cf. A000079, A000225, A000975, A004754, A004760, A052997, A054429, A080541, A081253, A081254, A266248, A266613, A266721, A267045.

PARI
```
See Links section.
```

a(n) = 1 iff n = A000225(k) for some k > 0.
a(n) = 2 iff n = A000079(k) for some k > 0.
a(n) = 5 iff n = A081254(k) for some k > 2.
a(n) = 6 iff n = A000975(k) for some k > 2.
a(n) = 10 iff n = A081253(k) for some k > 2.
a(n) = 12 iff n = A266613(k) for some k > 3.
a(n) = 13 iff n = A052997(k) for some k > 2.
a(n) = 14 iff n = A266721(k) for some k > 2.
a(n) = 18 iff n = A267045(k) for some k > 3.
a(n) = 54 iff n = A266248(k) for some k > 4.
These formulas come from the fact that each sequence on the right side, say f, eventually satisfies: f(n) = floor(f(n+1)/2), and f(n) and f(n+2) have the same parity.

cp's OEIS Frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

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A053646 Distance to nearest power of 2.

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A206374 a(n) = (7*4^n - 1)/3.

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A081252 Partial sums of A053646.

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A266071 Binary representation of the middle column of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

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A266613 Decimal representation of the middle column of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.

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A191665 Dispersion of A042963 (numbers >1, congruent to 1 or 2 mod 4), by antidiagonals.

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