A306468 Number of ways to write n as floor(i^2/3) + floor(j^2/3) + floor(k^2/3) with 1 <= i <= j <= k.

1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 2, 4, 3, 4, 2, 3, 4, 4, 2, 5, 4, 2, 5, 3, 6, 2, 3, 5, 6, 3, 4, 5, 5, 4, 3, 6, 4, 5, 5, 5, 5, 5, 2, 7, 6, 5, 5, 3, 6, 6, 4, 6, 8, 3, 6, 5, 7, 5, 3, 8, 6, 6, 5, 6, 8, 5, 4, 8, 6, 4, 7, 7, 6, 7, 2, 8, 10, 6, 6, 5, 7, 6

Offset

0,4

Comments

Farhi proved that a(n) > 0 for any n >= 0.

i^2/3 means (i^2)/3, of course, not i^(2/3). - N. J. A. Sloane, Feb 18 2019

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Bakir Farhi, An Elementary Proof that any Natural Number can be Written as the Sum of Three Terms of the Sequence floor(n^2/3), Journal of Integer Sequences, Vol. 17 (2014), #14.7.6.

Rémy Sigrist, PARI program for A306468

Example

For n = 42:
- let f(k) = floor(k^2/3),
- 42 can be written in 5 ways as f(i) + f(j) + f(k) with 1 <= i <= j <= k:
    f(1) + f(8) + f(8) = 0 + 21 + 21
    f(2) + f(2) + f(11) = 1 + 1 + 40
    f(2) + f(5) + f(10) = 1 + 8 + 33
    f(3) + f(6) + f(9) = 3 + 12 + 27
    f(4) + f(7) + f(8) = 5 + 16 + 21,
- hence a(42) = 5.

Prog

(PARI) See Links section.

Crossrefs

Cf. A000212.

Sequence in context: A353865 A237110 A078704 * A032358 A011960 A187035

Adjacent sequences: A306465 A306466 A306467 * A306469 A306470 A306471

Keyword

nonn

Author

Rémy Sigrist, Feb 17 2019

Status

approved