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Numbers having exactly one maximal group of consecutive zeros in binary representation of n.
5

%I #36 Apr 12 2019 08:27:18

%S 0,2,4,5,6,8,9,11,12,13,14,16,17,19,23,24,25,27,28,29,30,32,33,35,39,

%T 47,48,49,51,55,56,57,59,60,61,62,64,65,67,71,79,95,96,97,99,103,111,

%U 112,113,115,119,120,121,123,124,125,126,128,129,131,135,143,159,191

%N Numbers having exactly one maximal group of consecutive zeros in binary representation of n.

%C A087116(a(n)) = 1.

%C a(n) = A043687(n-1) for 1 < n < 1000. - _Georg Fischer_, Oct 19 2018

%H Gheorghe Coserea, <a href="/A087118/b087118.txt">Table of n, a(n) for n = 1..12343</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F From _Gheorghe Coserea_, Sep 28-30 2015: (Start)

%F a((n^3 - n)/6 + 2) = 2^n for n >= 1.

%F a((n^3 - n)/6 + 2 + n) = 2^n + 2^(n-1) for n >= 2.

%F a((n^3 - n)/6 + 2 + n + n-1) = 2^n + 2^(n-1) + 2^(n-2) for n >= 3.

%F a(n) < 2*2^((6*n)^(1/3)) and limsup a(n)/2^((6*n)^(1/3)) = 2.

%F a(n) > 1/2 * 2^((6*n)^(1/3)) for n>=3 and 1/2 <= liminf a(n)/(2^((6*n)^(1/3))) <= 1.

%F (End)

%p 0, seq(seq(seq(2^n - 2^b + 2^a - 1, a=0..b-1),b=n-1..1,-1),n=0..10); # _Robert Israel_, Oct 01 2015

%t Table[2^n - 2^b + 2^a - 1, {n, 0, 10}, {b, n-1, 1, -1}, {a, 0, b-1}] // Flatten // Prepend[#, 0]& (* _Jean-François Alcover_, Apr 11 2019, after _Robert Israel_ *)

%o (PARI)

%o num(a,b,c) = (1 << (a+b+c)) - (1 << (b+c)) + (1 << c) - 1;

%o succ(a,b,c) = {

%o if (b > 1, return([a, b-1, c+1]));

%o if (c > 0, return([a+1, c, 0]));

%o return([1, a+1, 0]);

%o };

%o seq(n) = {

%o my(a = 1, b = 1, c = 0, v = vector(n));

%o for (i = 2, n, v[i] = num(a,b,c);

%o my(x = succ(a,b,c)); a = x[1]; b = x[2]; c = x[3]);

%o return(v);

%o };

%o seq(64) \\ _Gheorghe Coserea_, Sep 28 2015

%Y Cf. A007088, A023416, A043687, A087119.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Aug 14 2003