A305233 Smallest k such that binomial(k, floor(k/2)) >= n.
1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 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, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 1
Keywords
References
- K. H. Kim, Boolean matrix theory and applications. Marcel Dekker, New York and Basel (1982).
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000
- Dmitry I. Ignatov and Yazag Meziane An asymptototic lower bound for A305233.
- Evgeny E. Marenich, Determining the Schein Rank of Boolean Matrices. Matrix Methods: Theory, Algorithms and Applications (2010) 85-103.
- Emanuel Sperner, Ein Satz über Untermengen einer endlichen Menge, Mathematische Zeitschrift 27, 544-548, (1928).
Programs
-
Mathematica
Array[Block[{k = 1}, While[Binomial[k, Floor[k/2]] < #, k++]; k] &, 105] (* Michael De Vlieger, Aug 02 2018 *) -
PARI
a(n) = my(k=1); while (binomial(k, floor(k/2)) < n, k++); k; \\ Michel Marcus, Jul 10 2018
-
PARI
first(n) = my(l=List(), t=1, res = vector(n), c=1); while(c<=n, listput(l,c); t++; c=binomial(t,t\2)); listput(l,c); res[1]=1; for(i=2, #l, m = max(n,l[i]); for(j=l[i-1] + 1, min(n,l[i]), res[j]=i)); res \\ David A. Corneth, May 22 2020
-
Python
from sympy import binomial def f(n): return binomial(n, n // 2) sum([[i]*(f(i)-f(i-1)) for i in range(1,10)],[1])
Formula
Asymptotic lower bound is log_2(sqrt(Pi/2)*n) + O(log_2(log_2(sqrt(Pi/2)*n))). - Dmitry I. Ignatov and Meziane Yazag, Dec 07 2023
Comments