A238454 Difference between 2^(2*n-1) and the next larger square.

2, 1, 4, 16, 17, 68, 89, 356, 697, 1337, 2449, 4001, 4417, 17668, 24329, 4633, 18532, 74128, 296512, 1186048, 1778369, 1181833, 4727332, 18909328, 28184177, 17830441, 71321764, 285287056, 381898097, 9092137, 36368548, 145474192, 581896768, 2327587072, 9310348288

Offset: 1

Author

Alex Ratushnyak, Feb 26 2014

Keywords

• nonn
• easy

Examples

			a(1) = 4 - 2^1 = 2.
a(2) = 9 - 2^3 = 1.
a(3) = 36 - 2^5 = 4.
		

Crossrefs

Cf. A000079, A056008, A201125, A236564.

Programs

Mathematica

(Floor[Sqrt[#]]+1)^2-#&/@Table[2^(2n-1),{n,40}] (* Harvey P. Dale, Jul 05 2019 *)

PARI

a(n) = my(r,s=sqrtint(1<<(2*n-1),&r)); 2*s+1-r; \\ Kevin Ryde, Oct 12 2022

Python

def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s:  sr = s
        b>>=1
    return sr
def a(n):
    nn = 2**(2*n+1)
    s = isqrt(nn)
    return (s+1)**2-nn
for n in range(77):  print(str(a(n)), end=',')

Sage

def a(n):
    return ceil(2^n/sqrt(2))^2 - 2^(2*n-1) # Ralf Stephan, Mar 08 2014

Formula

From Antti Karttunen, Feb 27 2014: (Start)

a(n) = ceiling(sqrt(2^(2*n-1)))^2 - 2^(2*n-1).

For all n, A000035(abs(A201125(n) - A238454(n))) = 1, because if the nearest square at the other side of 2^(2*n-1) is even, then the nearest square at the other side is odd.

(End)