A387269 Numbers whose binary expansion consists of alternating runs of 1's and 0's where each run of 0's is exactly one longer than the preceding run of 1's, and the expansion ends with a 0-run.

4, 24, 36, 112, 152, 196, 292, 480, 624, 792, 900, 1176, 1220, 1572, 1984, 2340, 2528, 3184, 3608, 3844, 4720, 4888, 4996, 6296, 6340, 7204, 8064, 9368, 9412, 9764, 10176, 12580, 12768, 14448, 15384, 15876, 18724, 18912, 19568, 19992, 20228, 25200, 25368, 25476

Offset

1,1

Comments

Every term is divisible by 4, since the final 0-run has length at least 2.

For k >= 1, the single-block 1^k 0^(k+1) = 2^(2k+1) - 2^(k+1) = A059153(k-1) is a term.

The function x -> x & (2*x) shortens each 1-run by 1 and lengthens the following 0-run by 1, mapping A387270 termwise onto this sequence.

Transforming each single-block 1^k 0^(k+1) to a digit k, generates a sequence like A069800, but in a different order: for example "5" comes before "1111".

Links

Ahmet Caglar Saygili, Table of n, a(n) for n = 1..10000

Sean A. Irvine, Java program (github)

Formula

a(n) = A213370(A387270(n)) = A387270(n) & (2*A387270(n)).

Example

36 = 100100_2 is a term since its pairs of (1 then 0) runs are (1,2), (1,2).

Mathematica

Select[Range[4, 30000, 4], AllTrue[Differences[Map[Length, Split[IntegerDigits[#, 2]]]][[;; ;; 2]], # == 1 &] &] (* Paolo Xausa, Oct 22 2025 *)

Prog

(Julia)
function ok(n::Integer)::Bool
    n > 0 && iseven(n) || return false
    x = unsigned(n)
    while x != 0
        z = trailing_zeros(x); x >>= z
        o = trailing_ones(x)
        z == o + 1 || return false
        x >>= o
    end
    true
end
[k for k in 0:10^5 if ok(k)]
(Python)
from itertools import groupby
def ok(n):
    L = [len(list(g)) for k, g in groupby(bin(n)[2:])]
    return (m:=len(L))&1 == 0 and all(L[2*j]+1 == L[2*j+1] for j in range(m>>1))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Aug 25 2025
(PARI) isok(k) = if (!(k%2), my(b=binary(k), pos=1, d, dd); for (i=1, #b-1, if (b[i] != b[i+1], if (b[i], d = i-pos+1; pos = i+1, dd = i-pos+1; pos = i+1; if (dd != d+1, return(0))))); dd = #b - pos+1; if (dd != d+1, return(0)); return(1); ); \\ Michel Marcus, Aug 26 2025
(PARI) isok(t)= my(ok=!!t, v); while(t, (ok=((v=valuation(t, 2))>1)) || break; (ok=(bitand(t, 4^v-1) == (2^(v-1)-1)<<v)) || break; t>>=(v*2-1)); ok; \\ Ruud H.G. van Tol, Apr 23 2026

Crossrefs

Cf. A059153, A166751, A387270, A213370.

Cf. A069800.

Sequence in context: A189228 A338786 A363218 * A048188 A121024 A120622

Adjacent sequences: A387266 A387267 A387268 * A387270 A387271 A387272

Keyword

base,nonn,easy

Author

Ahmet Caglar Saygili, Aug 24 2025

Status

approved