A231151 - OEIS

%I #8 Nov 16 2014 06:41:56

%S 2,2,2,3,3,4,4,4,5,5,6,6,7,7,7,8,8,9,9,9,10,10,11,11,11,12,12,13,13,

%T 14,14,14,15,15,16,16,16,17,17,18,18,19,19,19,20,20,21,21,21,22,22,23,

%U 23,23,24,24,25,25,26,26,26,27,27,28,28,28,29,29,30,30

%N Least integer k such that n+1 + ... + n+k > 1 + ... + n.

%C a(n) is the least k such that 2*p(n) < p(n+k), where p(n) is the n-th partial sum of the sequence c(n) = n.

%H Clark Kimberling, <a href="/A231151/b231151.txt">Table of n, a(n) for n = 2..1000</a>

%F a(n) = 1 + Floor[(-2 n - 1 + sqrt(8*n^2 + 8*n + 1)/2)].

%e 3 <= 1 + 2 < 3 + 4, so a(2) = 2

%e 4 <= 1 + 2 + 3 < 4 + 5, so a(3) = 3.

%e 11+12+13+14 <= 55 < 11+12+13+14+15, so a(10) = 5.

%t 1 + Table[test = n (1 + n); NestWhile[# + 1 &, n + 1, test >= -(n - #1) (1 + n + #1) &] - n - 1, {n, 2, 120}] (* _Peter J. C. Moses_, Nov 07 2013 *)

%t t = 1 + Table[Floor[(-2 n - 1 + Sqrt[8 n^2 + 8 n + 1])/2], {n, 2, 120}]

%Y Cf. A231152, A231153.

%K nonn,easy

%O 2,1

%A _Clark Kimberling_, Nov 09 2013