A350312 - OEIS

%I #28 Feb 18 2022 21:03:03

%S 0,1,10,11,101,110,111,1011,1101,1110,1111,10110,10111,11011,11101,

%T 11110,11111,101101,101110,101111,110110,110111,111011,111101,111110,

%U 111111,1011011,1011101,1011110,1011111,1101101,1101110,1101111,1110110,1110111,1111011

%N Narayana weighted representation of n (the bottom version). Also binary representation of numbers not containing 00 or 010 as a substring.

%C a(n) equals binary representation of m, if and only if A350311(m) = n and for all k < m, A350311(k) < n.

%H A.H.M. Smeets, <a href="/A350215/a350215.pdf">The design of greedy number representations</a>

%F Regular expression: 0|11*(0111*)*(0|01|011*)?.

%t q[n_] := SequenceCount[IntegerDigits[n, 2], #] & /@ {{0, 0}, {0, 1, 0}} == {0, 0}; bin[n_] := FromDigits[IntegerDigits[n, 2]]; bin /@ Select[Range[0, 120], q] (* _Amiram Eldar_, Jan 27 2022 *)

%o (Python) # first method (as from definition)

%o def A101(n):

%o     f0, f1, f2, r = 1, 1, 1, 0

%o     while n > 0:

%o         if n%2 == 1:

%o             r = r+f0

%o         n, f0, f1, f2 = n//2, f0+f2, f0, f1

%o     return r

%o n, a = 0, 0

%o while n < 36:

%o     if A101(a) == n:

%o         print(bin(a)[2:], end = ", ")

%o         n += 1

%o     a += 1

%o (Python) # second method (as from regular expression)

%o def test(n):

%o     s, i, n1 = bin(n)[2:], 0, 2

%o     while i < len(s):

%o         if s[i] == "0":

%o             if n1 < 2:

%o                 return 0

%o             n1 = 0

%o         else:

%o             n1 += 1

%o         i += 1

%o     return 1

%o n, a = 0, 0

%o while n < 36:

%o     if test(a):

%o         print(bin(a)[2:], end = ", ")

%o         n += 1

%o     a += 1

%Y Cf. A000930, A048715, A350215 (top version), A350311.

%Y Fibonacci representations: A014417 (Zeckendorf), A104326 (dual Zeckendorf).

%K nonn,base

%O 0,3

%A _A.H.M. Smeets_, Dec 24 2021