A111651 n appears 3n times.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Author

Jonathan Vos Post, Aug 12 2005

Keywords

• easy
• nonn

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000194, A002024, A008585, A111650, A111652.

Programs

Mathematica

Table[PadRight[{},3n,n],{n,10}]//Flatten (* Harvey P. Dale, Sep 15 2021 *)

PARI

{a(n)=if(n<1, 0, polcoeff( x/(1-x)*prod(k=1, n\3, (1-x^(3*k))^(-1)^k, 1+O(x^n)), n))} /* Michael Somos, Aug 31 2006 */

PARI

a(n) = sqrtint(24*n) \/ 6; \\ Kevin Ryde, Aug 31 2024

Python

from math import isqrt
def A111651(n): return isqrt((n<<3)//3)+1>>1 # Chai Wah Wu, Oct 05 2024

Formula

Expansion of (q/(1-q))psi(q^3) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Aug 31 2006

G.f.: x/(1-x)*Product_{k>0} (1-x^(3k))^((-1)^k).

a(n) = round(sqrt((2/3)*n)) = A002024(ceiling(n/3)). - Kevin Ryde, Aug 31 2024