A058034 Number of numbers whose cube root rounds to n.

1, 3, 12, 27, 49, 75, 108, 147, 193, 243, 300, 363, 433, 507, 588, 675, 769, 867, 972, 1083, 1201, 1323, 1452, 1587, 1729, 1875, 2028, 2187, 2353, 2523, 2700, 2883, 3073, 3267, 3468, 3675, 3889, 4107, 4332, 4563, 4801, 5043, 5292, 5547, 5809, 6075, 6348

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Formula

a(n) = 3n^2+1 if n == 0 (mod 4), 3n^2 otherwise.

a(n) = A033428(n)+A011765(n) = A034131(n)-A034131(n-1) = A219085(n)-A219085(n-1). - Corrected and extended by Zhuorui He, Dec 13 2025

a(n) = (1+(-1)^n+(-i)^n+i^n+12*n^2)/4 where i=sqrt(-1). - Colin Barker, Jul 04 2014

G.f.: -(3*x^5+6*x^4+6*x^3+7*x^2+x+1) / ((x-1)^3*(x+1)*(x^2+1)). - Colin Barker, Jul 04 2014

Example

a(2)=12 since the cube roots of 4, 5, 6, ..., 15 all lie between 1.5 and 2.5.

Maple

seq(1 + floor((n+1/2)^3) - ceil((n-1/2)^3), n = 0 .. 100);

Mathematica

Table[SeriesCoefficient[-(3 x^5 + 6 x^4 + 6 x^3 + 7 x^2 + x + 1)/((x - 1)^3 (x + 1) (x^2 + 1)), {x, 0, n}], {n, 0, 46}] (* Michael De Vlieger, Dec 24 2015 *)
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {1, 3, 12, 27, 49, 75}, 50] (* Vincenzo Librandi, Dec 25 2015 *)

Prog

(PARI) Vec(-(3*x^5+6*x^4+6*x^3+7*x^2+x+1)/((x-1)^3*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Jul 04 2014
(Magma) [n mod 4 eq 0 select 3*n^2+1 else 3*n^2: n in [0..80]]; // Vincenzo Librandi, Dec 25 2015

Crossrefs

Cf. A003215 (number of numbers whose floor (or ceiling) of the cube root is n), A004277 (number of numbers whose square root rounds to n).

First differences of A034131 and A219085.

Cf. A033428, A011765.

Sequence in context: A033428 A213486 A018230 * A294416 A187273 A361847

Adjacent sequences: A058031 A058032 A058033 * A058035 A058036 A058037

Keyword

easy,nonn

Author

Henry Bottomley, Nov 22 2000

Status

approved