A231149 Greatest integer k such that n+1 + ... + n+k <= 1 + ... + n.

1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29

Offset

2,4

Comments

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

Links

Clark Kimberling, Table of n, a(n) for n = 2..1000

Formula

a(n) = floor((-2*n - 1 + sqrt(8*n^2 + 8*n + 1)/2)).

Example

3 <= 1 + 2 < 3 + 4, so a(2) = 1.
4 <= 1 + 2 + 3 < 4 + 5, so a(3) = 1.
11+12+13+14 <= 55 < 11+12+13+14+15, so a(10) = 4.

Mathematica

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

Crossrefs

Cf. A231151, A231153.

Sequence in context: A054071 A028827 A083055 * A195072 A121828 A057357

Adjacent sequences: A231146 A231147 A231148 * A231150 A231151 A231152

Keyword

nonn,easy

Author

Clark Kimberling, Nov 09 2013

Status

approved