cp's OEIS Frontend

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/4 and { } = fractional part. Equivalently, the jump sequence of f(x) = x^(1/4), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p). For details and a guide to related sequences, see A219085.

-4*a(n) gives the real part of (n+n*i)*((n+1)+n*i)*(n+(n+1)*i)*((n+1)+(n+1)*i). The imaginary part is always zero. - Jon Perry, Feb 05 2014

Numbers k such that 16*k+1 is a fourth power. - Bruno Berselli, May 29 2018

The row sums of "Floyd's Triangle", which is a triangular array of natural numbers beginning with the number 1, produce the sequence A006003. A006003 can be bisected to get the Rhombic Dodecahedron Sequence A005917, whose n-th partial sum is n^4, and A317297, whose n-th partial sum is a(n). Interleave n^4 or A000583 back with {a(n)} to get A011863, whose first differences are A019298. Finally, A011863(n)-A011863(n-2) = A006003(n-1). - Bruce J. Nicholson, Dec 22 2019

Numerators in approximation sqrt(2) ~ a(n)/2^n.

a(n) is the number k such that {log_2(k)} < 1/2 < {log_2(k+1)}, where { } = fractional part. Equivalently, the jump sequence of f(x) = log_2(x), in the sense that these are the positive integers k for which round(log_2(k)) < round(log_2(k+1)); see A219085. - Clark Kimberling, Jan 01 2013

Largest k such that k^2 <= 2^(2n + 1). - Irina Gerasimova, Jul 07 2013

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 3/4 and { } = fractional part. Equivalently, the jump sequence of f(x) = x^(3/4), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p). For details and a guide to related sequences, see A219085.

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/5 and { } = fractional part. Equivalently, the jump sequence of f(x) = x^(1/5), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p). For details and a guide to related sequences, see A219085.

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/6 and { } = fractional part. Equivalently, the jump sequence of f(x) = x^(1/6), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p). For details and a guide to related sequences, see A219085.

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/7 and { } = fractional part. Equivalently, the jump sequence of f(x) = x^(1/7), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p). It appears that the sequence is linearly recurrent with order 71. For details and a guide to related sequences, see A219085.

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/8 and { } = fractional part. Equivalently, the jump sequence of f(x) = x^(1/8), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p). It appears that the sequence is linearly recurrent with order 23. Compare its signature with row 9 of the triangle at A008949. For which values of p is there a match of this sort between the jump sequence of x^p and row p+1 of the triangle?

For details and a guide to related sequences, see A219085.

a(n) is the number k such that {log(k)} < 1/2 < {log(k+1)}, where { } = fractional part. Equivalently, the jump sequence of f(x) = log(x), in the sense that these are the positive integers k for which round(log(k)) < round(log(k+1)). For a guide to related sequences, see A219085.