A278814 a(n) = ceiling(sqrt(3n+1)).

1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 0

Author

Mohammad K. Azarian, Nov 28 2016

Keywords

• nonn
• easy

Crossrefs

Cf. A002162, A016777, A016789, A016933, A017569, A032765, A058183, A131033, A007494, A051536, A007559.

Programs

Derive

PROG(y := [], n := 100, LOOP(IF(n = -1, RETURN y), y := ADJOIN(CEILING(SQRT(1 + 3·n)), y), n := n - 1))

Maple

seq(ceil(sqrt(3*k+1)), k=0..100); # Robert Israel, Nov 28 2016

Mathematica

Table[Ceiling[Sqrt[3n+1]],{n,0,100}]

PARI

a(n)=sqrtint(3*n)+1 \\ Charles R Greathouse IV, Nov 29 2016

Python

from math import isqrt
def A278814(n): return 1+isqrt(3*n) # Chai Wah Wu, Jul 28 2022

Formula

a(n) = ceiling(sqrt(3n+1)).

From Robert Israel, Nov 28 2016: (Start)

G.f.: (1-x)^(-1)*Sum_{k>=0} (x^(3*k^2)+x^(3*k^2+2*k+1)+x^(3*k^2+4*k+2)).

a(n+1) = a(n)+1 if n is in A032765, otherwise a(n+1) = a(n). (End)

Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Jun 18 2025