A097337 - OEIS

%I #21 Nov 01 2021 08:15:05

%S 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,

%T 16,17,17,18,19,19,20,20,21,21,22,23,23,24,24,25,25,26,27,27,28,28,29,

%U 30,30,31,31,32,32,33,34,34,35,35,36,36,37,38,38,39,39,40,40,41,42,42,43

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

%C 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.

%C The integers k such that a(k) = a(k+1) give A054406. - _Michel Marcus_, Nov 01 2021

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

%H Karl V. Keller, Jr., <a href="/A097337/b097337.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F 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).

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

%o (PARI) a(n)=sqrtint(n^2\3) \\ _Charles R Greathouse IV_, Nov 01 2021

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

%K nonn,easy

%O 1,4

%A _Cino Hilliard_, Sep 17 2004