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

1,4

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A340763 A286103 A056556 * A151982 A259361 A111854

Adjacent sequences: A111648 A111649 A111650 * A111652 A111653 A111654

Keyword

easy,nonn

Author

Jonathan Vos Post, Aug 12 2005

Status

approved