A236564 - OEIS

%I #44 May 05 2021 13:40:07

%S 1,-1,-4,7,-17,23,-89,7,28,112,448,1792,-4417,5503,22012,-4633,-18532,

%T -74128,-296512,296863,1187452,-1181833,-4727332,4817239,19268956,

%U -17830441,-71321764,94338007,377352028,-9092137,-36368548,-145474192,-581896768,-2327587072,-9310348288

%N Difference between 2^(2n-1) and the nearest square.

%C The distances of the even powers 2^(2n) to their nearest squares are obviously all zero and therefore skipped.

%H Vincenzo Librandi, <a href="/A236564/b236564.txt">Table of n, a(n) for n = 1..500</a>

%F If A201125(n) < A238454(n), a(n) = A201125(n), otherwise a(n) = -A238454(n). [Negative terms are for cases where the nearest square is above 2^(2n-1), not below it.] - _Antti Karttunen_, Feb 27 2014

%e a(1) = 2^1 - 1^2 = 1.

%e a(2) = 2^3 - 3^2 = -1.

%e a(3) = 2^5 - 6^2 = 32 - 36 = -4.

%p A236564 := proc(n)

%p     local x,sq,lo,hi ;

%p     x := 2^(2*n-1) ;

%p     sq := isqrt(x) ;

%p     lo := sq^2 ;

%p     hi := (sq+1)^2 ;

%p     if abs(x-lo) < abs(x-hi) then

%p         x-lo ;

%p     else

%p         x-hi ;

%p     end if;

%p end proc: # _R. J. Mathar_, Mar 13 2014

%t Table[2^n - Round[Sqrt[2^n]]^2, {n, 1, 79, 2}] (* _Alonso del Arte_, Feb 23 2014 *)

%o (Python)

%o def isqrt(a):

%o     sr = 1 << (int.bit_length(int(a)) >> 1)

%o     while a < sr*sr:  sr>>=1

%o     b = sr>>1

%o     while b:

%o         s = sr + b

%o         if a >= s*s:  sr = s

%o         b>>=1

%o     return sr

%o for n in range(47):

%o     nn = 2**(2*n+1)

%o     a = isqrt(nn)

%o     d1 = nn - a*a

%o     d2 = (a+1)**2 - nn

%o     if d2 < d1:  d1 = -d2

%o     print(str(d1), end=',')

%Y Cf. A053188, A201125, A238454.

%K sign,easy

%O 1,3

%A _Alex Ratushnyak_, Feb 23 2014