A043320 Numbers which, written in base 256, have all digits less than 16 and no two adjacent digits equal.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 256, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 512, 513, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 768, 769, 770, 772, 773, 774

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

Sequence A033014 consists of the numbers that have all base 16 digits repeated *exactly* twice. (This is equivalent to say that the base-256 digits are 0x00, 0x11, 0x22,... or 0xFF, in hex notation, and no two adjacent base-256 digits are equal.) Thus, these numbers are divisible by 0x11 = 17, and the result of the division is a number which has no other base-256 digits than 0x00, 0x01,... or 0x0F, and no two adjacent digits equal. Conversely, it is clear that exactly these numbers are terms of A033014 when multiplied by 17 = 0x11. - M. F. Hasler, Feb 05 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1800

Crossrefs

Cf. A043307 - A043319, A043291, A033001 - A033014, A033015 - A033029.

Programs

Mathematica

Select[Range[20000], Union[Length/@Split[IntegerDigits[#, 16]]]=={2}&]/17 (* Vincenzo Librandi, Feb 06 2014 *)

PARI

is_A043320(n)={(n=[n])&&!until(!n[1], ((n=divrem(n[1], 256))[2]<16 && n[1]%16!=n[2])||return)} \\ M. F. Hasler, Feb 03 2014

Python

from itertools import count, islice, groupby
def A043320_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:set(len(list(g)) for k, g in groupby(hex(17*n)[2:]))=={2},count(max(startvalue,1)))
A043320_list = list(islice(A043320_gen(),20)) # Chai Wah Wu, Mar 10 2023

Formula

a(n) = A033014(n)/17. [This was initially the definition of the sequence. - M. F. Hasler, Feb 03 2014]

Extensions

New definition by M. F. Hasler, Feb 03 2014