A235963 n appears (n+1)/(1 + (n mod 2)) times.

0, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Author

Mircea Merca, Jan 17 2014

Keywords

• nonn
• tabf

Comments

n appears A001318(n+1) - A001318(n) = A026741(n+1) times.

Sum_{k=0...a(n)} (-1)^ceiling(k/2)*p(n-G(k)) = 0 for n>0, where p(n)=A000041(n) is the partition function, and G(k)=A001318(k) denotes the generalized pentagonal numbers.

Row lengths of A238442, n >= 1. - Omar E. Pol, Dec 22 2016

Examples

			As a triangle:
  0;
  1;
  2, 2, 2;
  3, 3;
  4, 4, 4, 4, 4;
  5, 5, 5;
  6, 6, 6, 6, 6, 6, 6;
  7, 7, 7, 7;
  8, 8, 8, 8, 8, 8, 8, 8, 8;
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

First differences are A080995.

Cf. A000041, A001318, A026741, A085141, A180447, A238442.

Programs

Maple

T:= n-> n$(n+1)/(n mod 2+1):
seq(T(n), n=0..13);  # Alois P. Heinz, Nov 23 2024

Mathematica

Table[Table[n, {(n + 1)/(1 + Mod[n, 2])}], {n, 0, 14}]//Flatten (* T. D. Noe, Jan 29 2014 *)

Python

from math import isqrt
def A235963(n): return (m:=isqrt((n+1<<3)//3))-(n+1<=(m*(3*m+4)+1 if m&1 else m*(3*m+2))>>3) # Chai Wah Wu, Nov 23 2024

Python

A235963=lambda n: ((s:=isqrt(24*n+1))+1)//6+(s-1)//6 # Natalia L. Skirrow, May 13 2025

Formula

Let t = (sqrt(n*8/3 + 1) - 1)/2 + 1/3 and let k = floor(t); then a(n) = 2k if t - k < 2/3, 2k+1 otherwise. - Jon E. Schoenfield, Jun 13 2017

a(n) = m if n+1>A001318(m) and a(n) = m-1 otherwise where m = floor(sqrt(8(n+1)/3)). - Chai Wah Wu, Nov 23 2024

From Natalia L. Skirrow, May 13 2025: (Start)

a(n) = A180447(n) + A085141(n).

a(n) = floor((s+1)/6) + floor((s-1)/6) where s=floor(sqrt(24*n+1)).

G.f.: (f(x,x^2)-1)/(1-x), where f is Ramanujan's bivariate theta function. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + (2-sqrt(2))*log(2)/8 + log(2+sqrt(2))/(2*sqrt(2)). - Amiram Eldar, May 28 2025