A097337 Integer part of the edge of a cube that has space-diagonal n.

0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43

Offset: 1

Author

Cino Hilliard, Sep 17 2004

Keywords

• nonn
• easy

Comments

The first few terms are the same as A038128. However, A038128 is generated by Euler's constant = 0.5772156649015328606065120901..., which is close but not equal to 1/sqrt(3) = 0.5773502691896257645091487805..., which generates this sequence. Euler/(1/sqrt(3)) = 0.9997668585341064519813571911... and the equality fails in the 97th term.

The integers k such that a(k) = a(k+1) give A054406. - Michel Marcus, Nov 01 2021

References

• The Universal Encyclopedia of Mathematics, English translation, 1964, p. 155.

Links

• Karl V. Keller, Jr., Table of n, a(n) for n = 1..1000
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A020760 (1/sqrt(3)), A054406.

Programs

PARI

f(n) = for(x=1,n,s=x\sqrt(3);print1(s","));s

PARI

a(n)=sqrtint(n^2\3) \\ Charles R Greathouse IV, Nov 01 2021

Formula

Let L be the length of the edges. Then sqrt(2)*L is the diagonal of a face. Whence n^2 = 2*L^2 + L^2, or n = sqrt(3)*L and L = n/sqrt(3).