A380358 Numbers whose binary expansion ends with 11 and does not contain adjacent zeros.

3, 7, 11, 15, 23, 27, 31, 43, 47, 55, 59, 63, 87, 91, 95, 107, 111, 119, 123, 127, 171, 175, 183, 187, 191, 215, 219, 223, 235, 239, 247, 251, 255, 343, 347, 351, 363, 367, 375, 379, 383, 427, 431, 439, 443, 447, 471, 475, 479, 491, 495, 503, 507, 511, 683

Offset: 1

Author

R. J. Cintra, Jan 22 2025

Keywords

• nonn
• base

Comments

The numbers in this sequence appear in the conversion of conventional binary numbers to the canonical signed-digit representation.

Examples

			183 is in the sequence because its binary expansion is 10110111.
		

References

• J. L. Smith and A. Weinberger, "Shortcut Multiplication for Binary Digital Computers", in Methods for High-Speed Addition and Multiplication, National Bureau of Standards Circular 591, Sec. 1, February, 1958, page 21.

Links

• R. J. Cintra, A note on the conversion of nonnegative integers to the canonical signed-digit representation, arXiv:2501.10908 [eess.SP], 2025.

Crossrefs

Cf. A003754, A052499, A247648, A365808, A365809.

Programs

Mathematica

Select[4*Range[0, 170] + 3, SequencePosition[IntegerDigits[#, 2], {0, 0}] == {} &] (* Amiram Eldar, Feb 05 2025 *)

Python

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

Formula

a(n) = 2 * A247648(n) + 1.

From Hugo Pfoertner, Feb 07 2025: (Start)

a(n) = 4*A052499(n) - 1.

a(n) = 4*(A365808(n+1) + 1)/3 - 1.

a(n) = 2*(A365809(n) + 1)/3 - 1. (End)