A267705 -id:A267705 - cp's OEIS frontend

A357774 Binary expansions of odd numbers with two zeros in their binary expansion.

1001, 10011, 10101, 11001, 100111, 101011, 101101, 110011, 110101, 111001, 1001111, 1010111, 1011011, 1011101, 1100111, 1101011, 1101101, 1110011, 1110101, 1111001, 10011111, 10101111, 10110111, 10111011, 10111101, 11001111, 11010111, 11011011, 11011101, 11100111, 11101011

Offset: 1

Bernard Schott, Oct 19 2022

For m >= 4, there are A000217(m-3) terms with m digits.

Cf. A000217, A007088, A357773.
A267524 \ {1, 10, 100} and A267705 \ {1, 10} are two subsequences.
Similar, but with k zeros in their binary expansion: A000042 (k=0), A190619 (k=1).

FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[1, 250, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 19 2022 *)

isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ A357773
f(n) = fromdigits(binary(n), 10); \\ A007088
lista(nn) = apply(f, select(isok, [1..nn])); \\ Michel Marcus, Oct 19 2022

from itertools import combinations, count, islice
def agen(): # generator of terms
    for d in count(4):
        b, c = 2**d - 1, 2**(d-1)
        for i, j in combinations(range(1, d-1), 2):
            yield int(bin(b - (c >> i) - (c >> j))[2:])
print(list(islice(agen(), 30))) # Michael S. Branicky, Oct 19 2022

from itertools import count, islice
def A357774_gen(): # generator of terms
    for l in count(2):
        m = (10**(l+2)-1)//9
        for i in range(l,0,-1):
            k = m-10**i
            yield from (k-10**j for j in range(i-1,0,-1))
A357774_list = list(islice(A357774_gen(),30)) # Chai Wah Wu, Feb 19 2023

from math import isqrt, comb
from sympy import integer_nthroot
def A357774(n):
    a = (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2,3))+3
    b = isqrt((j:=comb(a-1,3)-n+1)<<3)+3>>1
    c = j-comb((r:=isqrt(w:=j<<1))+(w>r*(r+1)),2)
    return (10**a-1)//9-10**b-10**c # Chai Wah Wu, Dec 19 2024

a(n) = A007088(A357773(n)).

A188530 2^(2n+1)-5*2^(n-1)-1.

Original entry on oeis.org

2, 21, 107, 471, 1967, 8031, 32447, 130431, 523007, 2094591, 8383487, 33544191, 134197247, 536829951, 2147401727, 8589770751, 34359410687, 137438298111, 549754503167, 2199020634111

Offset: 1

Brad Clardy, Apr 03 2011

Starting with n=2, binary palindromic numbers of the form (n-1)010(n-1) where n is the index and the number of 1's

			first 6 term in binary starting with n=2 are 10101,1101011,111010111,11110101111,1111101011111,111111010111111

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A267705.

Table[2^(2n+1)-5 2^(n-1)-1,{n,20}] (* or *) Rest[CoefficientList[ Series[(x(-2-7x+12x^2))/((x-1)(2x-1)(4x-1)), {x,0,20}], x]]  (* Harvey P. Dale, Apr 19 2011 *)

print([2*4**n - 5*2**(n-1) - 1 for n in range(1, 50)]) # Karl V. Keller, Jr., Jun 09 2022

a(n) = 2^(2n+1)-2^(n+1)-2^(n-1)-1.
A052539(n) = a(n)-2*a(n-1) for n>1.
a(n)= +7*a(n-1) -14*a(n-2) +8*a(n-3). G.f. ( x*(-2-7*x+12*x^2) ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Apr 04 2011
a(n) = 2*4^n - 5*2^(n-1) - 1. - Karl V. Keller, Jr., Jun 09 2022

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A357774 Binary expansions of odd numbers with two zeros in their binary expansion.

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A188530 2^(2n+1)-5*2^(n-1)-1.

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