A374446 - OEIS

%I #9 Jul 13 2024 02:44:31

%S 2,11,14,47,50,59,62,191,194,203,206,239,242,251,254,767,770,779,782,

%T 815,818,827,830,959,962,971,974,1007,1010,1019,1022,3071,3074,3083,

%U 3086,3119,3122,3131,3134,3263,3266,3275,3278,3311,3314,3323,3326,3839,3842,3851,3854,3887,3890,3899

%N Positions of zeros in the expansion of Product_{k>=0} (1 - x^(2^k))^3; A373308(a(n)) = 0 for n >= 1.

%C Conjecture: a(n) = A001196(n) - 1 for n >= 1, where A001196 lists numbers with only even length runs in their binary expansion.

%H Paul D. Hanna, <a href="/A374446/b374446.txt">Table of n, a(n) for n = 1..511</a>

%e Product_{k>=0} (1 - x^(2^k))^3 = 1 - 3*x + 0*x^2 + 8*x^3 - 9*x^4 + 3*x^5 + 8*x^6 - 24*x^7 + 15*x^8 + 19*x^9 - 24*x^10 + 0*x^11 + 17*x^12 - 27*x^13 + 0*x^14 + 64*x^15 + ... + A373308(n)*x^n + ...

%e in which the coefficients of {x^2, x^11, x^14, x^47, ..., x^a(n), ...} are zero.

%e Compare to numbers with only even length runs in their binary expansion: A001196 = [3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255, 768, ...]; it appears that a(n) = A001196(n) - 1 for n >= 1.

%Y Cf. A001196, A000695, A373308.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jul 08 2024