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

0,3

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

Links

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

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

Example

As 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;
  ...

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 *)

Prog

(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

Crossrefs

Cf. A000041, A001318, A026741, A238442.

Sequence in context: A367128 A028829 A130855 * A100196 A278078 A094708

Adjacent sequences: A235960 A235961 A235962 * A235964 A235965 A235966

Keyword

nonn,tabf

Author

Mircea Merca, Jan 17 2014

Status

approved