A111652 -id:A111652 - cp's OEIS frontend

A111650 2n appears n times (n>0).

2, 4, 4, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 18, 18, 18, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Jonathan Vos Post, Aug 12 2005

Seen as a triangle read by rows: T(n,k) = 2*n, 1<=k<=n. - Reinhard Zumkeller, Mar 18 2011

Reinhard Zumkeller, Rows n = 1..128 of triangle, flattened

Cf. A000194, A111651, A111652.

Cf. A001650, A193832, A002024.

a111650 n k = a111650_tabl !! (n-1) !! (k-1)
a111650_row n = a111650_tabl !! (n-1)
a111650_tabl = iterate (\xs@(x:_) -> map (+ 2) (x:xs)) [2]
a111650_list = concat a111650_tabl
-- Reinhard Zumkeller, Nov 14 2015, Mar 18 2011

Table[Table[2n,n],{n,12}]//Flatten (* Harvey P. Dale, Apr 21 2018 *)

from math import isqrt
def A111650(n): return isqrt(n<<3)+1&-2 # Chai Wah Wu, Jun 06 2025

a(n) = 2*A002024(n). - Chai Wah Wu, Jun 06 2025

A111651 n appears 3n times.

Original entry on oeis.org

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

Jonathan Vos Post, Aug 12 2005

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).

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

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

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

{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 */

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

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

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

cp's OEIS Frontend

A111650 2n appears n times (n>0).

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