%I
%S 4,12,16,20,28,36,44,48,52,60,64,68,76,80,84,92,100,108,112,116,124,
%T 132,140,144,148,156,164,172,176,180,188,192,196,204,208,212,220,228,
%U 236,240,244,252,256,260,268,272,276,284,292,300,304,308,316,320,324,332
%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[200], 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.
%K nonn,easy
%O 1,1
%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
