This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The binary indices of 100 are {3,6,7}, with maximal runs ((3),(6,7)), with reversed lengths (2,1), which is the 5th composition in standard order, so a(100) = 5.
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Reverse[Length/@Split[bpe[n],#2==#1+1&]]],{n,0,100}]
a247875 n = a247875_list !! (n-1) a247875_list = filter (\x -> even x || f x) [0..] where f x = x > 0 && (x `mod` 4 == 0 || f (x `div` 2))
Module[{upto=100,odds},odds=Select[Range[1,upto,2], SequenceCount[ IntegerDigits[#,2],{0,0}]>0&];Sort[Join[odds,Range[0,upto,2]]]] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Aug 30 2015 *)
A247875_list = [n for n in range(10**3) if not n % 2 or '00' in bin(n)] # Chai Wah Wu, Sep 26 2014
. 2*21=42 | 101010 2*6=12 | 1100 . 21 | 10101 6 | 110 . -----------+------- ----------+----- . 21 IMPL 42 | 101010 -> a(21) = 42 6 IMPL 12 | 1101 -> a(6) = 13 .
a265716 n = n `bimpl` (2 * n) where
bimpl 0 0 = 0
bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
where (p', u) = divMod p 2; (q', v) = divMod q 2
A265716 := n -> Bits:-Implies(n, 2*n): seq(A265716(n), n=0..61); # Peter Luschny, Sep 23 2019
IMPL[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[k, 2]]-1-n, k]];
a[n_] := n ~IMPL~ (2n);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 16 2021 *)
a(n)=bitor(bitneg(n, exponent(n)+1), 2*n) \\ Charles R Greathouse IV, Jan 20 2023
Here is the neighborhood: [0, 0, X, 0, 0] [0, 0, X, 0, 0] [X, X, X, X, X] [0, 0, X, 0, 0] [0, 0, X, 0, 0] which contains a(1) = 9 ON cells. The second and third generations are: [0, 0, 0, 0, X, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, X, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 0] [X, 0, X, 0, X, 0, X, 0, X] (again with 9 ON cells) [0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, X, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, X, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, X, X, 0, X, X, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, X, 0, 0, X, X, X, 0, 0, X, 0, 0] [0, 0, X, 0, X, 0, 0, 0, X, 0, X, 0, 0] [X, X, 0, 0, X, 0, X, 0, X, 0, 0, X, X] (with 37 ON cells) [0, 0, X, 0, X, 0, 0, 0, X, 0, X, 0, 0] [0, 0, X, 0, 0, X, X, X, 0, 0, X, 0, 0] [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, X, X, 0, X, X, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0] [0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0] The terms can be arranged into blocks of sizes 1,1,2,4,8,16,32,...: 1, 9, 9, 37, 9, 65, 37, 157, 9, 81, 65, 237, 37, 293, 157, 713, 9, 81, 81, 333, 65, 473, 237, 1077, 37, 333, 293, 1129, 157, 1285, 713, 2737, 9, 81, 81, 333, 81, 585, 333, 1413, 65, 585, 473, 1733, 237, 1933, 1077, 4337, 37, 333, 333, 1369, 293, 2125, 1129, 4969, 157, 1413, 1285, 5041, 713, 5561, 2737, 11421, ... The final terms in the rows are A246315.
C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/x^2+1/x+1+x+x^2+1/y^2+1/y+y+y^2;
OddCA(f, 70);
c[f_] := f /. {x -> 1, y -> 1};
OddCA[f_, M_] := Module[{a = {}, f2, p = 1}, f2 = PolynomialMod[f, 2]; Do[ AppendTo[a, c[p]]; Print[a]; p = PolynomialMod[p f2, 2], {n, 0, M}]; a];
f = 1/x^2 + 1/x + 1 + x + x^2 + 1/y^2 + 1/y + y + y^2;
OddCA[f, 70] (* Jean-François Alcover, May 24 2020, after Maple *)
import sympy
from operator import mul
from functools import reduce
x, y = sympy.symbols('x y')
f = ((1/x**2+1/x+1+x+x**2)*(1/y**2+1/y+1+y+y**2)).expand(modulus=2)
A247650_list, g = [1], 1
for n in range(1, 101):
s = [int(d, 2) for d in bin(n)[2:].split('00') if d != '']
g = (g*f).expand(modulus=2)
if len(s) == 1:
A247650_list.append(g.subs([(x, 1), (y, 1)]))
else:
A247650_list.append(reduce(mul, (A247650_list[d] for d in s)))
# Chai Wah Wu, Sep 25 2014
a:= n-> Bits[Nor](n, 2*n): seq(a(n), n=0..127);
a(n) = my(x=bitor(n, 2*n)); bitneg(x, #binary(x)); \\ Michel Marcus, Apr 14 2020
def A334076(n): m = n|(2*n) return 0 if n == 0 else 2**(len(bin(m))-2)-1-m # Chai Wah Wu, Apr 14 2020
First few rows are: 1; 3, 5; 7, 13, 11, 21; 15, 29, 27, 53, 23, 45, 43, 85; 31, 61, 59, 117, 55, 109, 107, 213, 47, 93, 91, 181, 87, 173, 171, 341;
T(n, k) = if(n==0, 1, 2*(k%2+1)*T(n-1, k\2)+1);
For n=5, the terms of A247648 that are in the interval [16, 31] are 21, 23, 27, 29, and 31, so a(5) = 21+23+27+29+31 = 131.
LinearRecurrence[{3, 3, -6, -4}, {1, 3, 12, 39}, 30] (* Paolo Xausa, Jun 19 2024 *)
Vec(x/((1 - x - x^2)*(1 - 2*x - 4*x^2)) + O(x^40)) \\ Michel Marcus, Jun 16 2024
183 is in the sequence because its binary expansion is 10110111.
Select[4*Range[0, 170] + 3, SequencePosition[IntegerDigits[#, 2], {0, 0}] == {} &] (* Amiram Eldar, Feb 05 2025 *)
from itertools import count, islice def A380358_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda n:n&3==3 and not '00' in bin(n),count(max(startvalue,1))) A380358_list = list(islice(A380358_gen(),20)) # Chai Wah Wu, Feb 12 2025
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