%N Numbers of the form (2*m - 1)*4^k where m >= 1, k >= 1.
%C Numbers of terms in nonnegative integer sequences the sum of which is never a square.
%C The sum of a sequence of consecutive nonnegative integers starting with k is never a square for any k, if and only if the number of the terms in the sequence can be expressed as (2*m - 1) * 2^(2*n), m and n being any positive integers. (Proved by Alfred Vella, Jun 14 2005.)
%C Odious and evil terms alternate. - _Vladimir Shevelev_, Jun 22 2009
%C Even numbers whose binary representation ends in an even number of zeros. - _Amiram Eldar_, Jan 12 2021
%H Amiram Eldar, <a href="/A108269/b108269.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 6*n + O(log n). - _Charles R Greathouse IV_, Nov 03 2016 [Corrected by _Amiram Eldar_, Jan 12 2021]
%F a(n) = 2 * A036554(n) = 4 * A003159(n). - _Amiram Eldar_, Jan 12 2021
%e a( 1, 1 ) = 4, a( 2, 1) = 12, etc.
%e For a( 1, 1 ): the sum of 4 consecutive nonnegative integers (4k+6, if the first term is k) is never a square.
%t Select[2 * Range, EvenQ @ IntegerExponent[#, 2] &] (* _Amiram Eldar_, Jan 12 2021 *)
%o (PARI) is(n)=my(e=valuation(n,2)); e>1 && e%2==0 \\ _Charles R Greathouse IV_, Nov 03 2016
%Y Intersection of A005843 and A003159.
%Y Cf. A000069, A001969, A036554.
%A Andras Erszegi (erszegi.andras(AT)chello.hu), May 30 2005
%E Entry revised by _N. J. A. Sloane_, Jun 26 2005
%E More terms from _Amiram Eldar_, Jan 12 2021