A133457 -id:A133457 - cp's OEIS frontend

A372325 Numbers whose binary expansion has an even number of 1's among positions listed in this sequence.

0, 2, 5, 7, 8, 10, 13, 15, 16, 18, 21, 23, 24, 26, 29, 31, 33, 35, 36, 38, 41, 43, 44, 46, 49, 51, 52, 54, 57, 59, 60, 62, 64, 66, 69, 71, 72, 74, 77, 79, 80, 82, 85, 87, 88, 90, 93, 95, 97, 99, 100, 102, 105, 107, 108, 110, 113, 115, 116, 118, 121, 123, 124

Offset: 1

David A. Madore, Apr 27 2024

			118 is in the sequence because 118 = 2^6 + 2^5 + 2^4 + 2^2 + 2^1, and an even number of the exponents 6,5,4,2,1 (namely 2,5) are in the sequence.
8192 is not in the sequence because 8192 = 2^13, and 13 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A133457, A272011.

R:= 0: RL:= [1]: nextp:= 2: m:= 1: count:= 0:
for i from 1 while count < 100 do
  L:= convert(i,base,2);
  if i = nextp then
    nextp:= 2*nextp;
    if R[1+nops(RL)] = m then RL:= [op(RL),m+1] fi;
    m:= m+1;
  fi;
  if convert(L[RL],`+`)::even
  then R:= R,i; count:= count+1
  fi
od:
R; # Robert Israel, May 28 2024

from itertools import count, islice
def agen():  # generator of terms
    aset = 0 # stored as a bitmask
    for k in count(0):
        if (k&aset).bit_count()%2 == 0:
            yield k
            aset += (1<Michael S. Branicky, Apr 28 2024

A348175 Irregular table T(n,k) read by rows: T(n,k) = T(n-1,k/2) when k is even and 3*T(n-1,(k-1)/2) + 2^(n-1) when k is odd. T(0,0) = 0 and 0 <= k <= 2^n-1.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 5, 0, 4, 2, 10, 1, 7, 5, 19, 0, 8, 4, 20, 2, 14, 10, 38, 1, 11, 7, 29, 5, 23, 19, 65, 0, 16, 8, 40, 4, 28, 20, 76, 2, 22, 14, 58, 10, 46, 38, 130, 1, 19, 11, 49, 7, 37, 29, 103, 5, 31, 23, 85, 19, 73, 65, 211

Offset: 0

Ryan Brooks, Oct 04 2021

			n\k 0  1  2  3  4  5  6  7
0   0
1   0  1
2   0  2  1  5
3   0  4  2 10  1  7  5 19

Cf. A001047 (right diagonal), A002697 (row sums), A119733.

Cf. A133457 (binary exponents).

T[0, 0] = 0; T[n_, k_] := T[n, k] = If[EvenQ[k], T[n - 1, k/2], 3*T[n - 1, (k - 1)/2] + 2^(n - 1)]; Table[T[n, k], {n, 0, 5}, {k, 0, 2^n - 1}] // Flatten (* Amiram Eldar, Oct 11 2021 *)

T(n, k) = if ((n==0) && (k==0), 0, if (k%2, 3*T(n-1,(k-1)/2) + 2^(n-1), T(n-1,k/2)));
tabf(nn) = for (n=0, nn, for (k=0, 2^n-1, print1(T(n,k), ", ")); print); \\ Michel Marcus, Oct 18 2021

T(n,k) = my(ret=0); for(i=0,n-1, if(bittest(k,n-1-i), ret=3*ret+1<Kevin Ryde, Oct 22 2021

T(n,k) = T(n-1,k/2) for k being even.
T(n,k) = 3*T(n-1,(k-1)/2) + 2^(n-1) for k being odd.
T(n,k) = 2*T(n-1,k) for 0 <= k <= 2^(n-1) - 1.
T(n,k) = Sum_{i=0..r} 2^(n-1-e[i]) * 3^i where binary expansion k = 2^e[0] + 2^e[1] + ... + 2^e[r] with ascending e[0] < e[1] < ... < e[r] (A133457). - Kevin Ryde, Oct 22 2021

A365396 Inverse permutation to A243571.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 13, 11, 14, 15, 21, 12, 16, 17, 22, 18, 23, 24, 34, 19, 25, 26, 35, 27, 36, 37, 55, 20, 28, 29, 38, 30, 39, 40, 56, 31, 41, 42, 57, 43, 58, 59, 89, 32, 44, 45, 60, 46, 61, 62, 90, 47, 63, 64, 91, 65, 92, 93, 144, 33, 48, 49, 66

Offset: 1

Alexander Khudyakov, Sep 03 2023

nonn

Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A000120, A000523, A056792, A133457, A243571.

a(n) = my(L = logint(n, 2), wt = hammingweight(n), A = L + wt, m = 0); fibonacci(A+1) + sum(i=1, L, binomial(i-1, A-i)) + sum(j=0, L-1, if(bittest(n, j), m++; binomial(j, m)))

a(n) = A000045(b(n) + 1) + Sum_{i=1..A000523(n)} binomial(i-1, b(n) - i) + Sum_{j=1..A000120(n) - 1} binomial(A133457(n, j), j) where b(n) = A056792(n).

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A372325 Numbers whose binary expansion has an even number of 1's among positions listed in this sequence.

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A348175 Irregular table T(n,k) read by rows: T(n,k) = T(n-1,k/2) when k is even and 3*T(n-1,(k-1)/2) + 2^(n-1) when k is odd. T(0,0) = 0 and 0 <= k <= 2^n-1.

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A365396 Inverse permutation to A243571.

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