A003057 n appears n - 1 times.

2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 2, 1 <= k <= n - 1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002

Smallest integer such that n-1 <= C(a(n),2). - Frank Ruskey, Nov 06 2007

a(n) = inverse (frequency distribution) sequence of A161680. - Jaroslav Krizek, Jun 19 2009

Taken as a triangle t(n, m) with offset 1, i.e., n >= m >= 1, this gives all positive integer limits r = r (a = m, b = A063929(n, m)) of the (a,b)-Fibonacci ratio F(a,b;k+1)/F(a,b;k) for k -> infinity. See the Jan 11 2015 comment on A063929. - Wolfdieter Lang, Jan 12 2015

Square array, T(n,k) = n + k + 2, n > = 0 and k >= 0, read by antidiagonals. Northwest corner:

2, 3, 4, 5, ...

3, 4, 5, 6, ...

4, 5, 6, 7, ...

5, 6, 7, 8, ...

... - Franck Maminirina Ramaharo, Nov 21 2018

a(n) is the pair chromatic number of an empty graph with n vertices. (The pair chromatic number of a graph G is the smallest number of colors for which G has a coloring where every vertex has two distinct colors, no adjacent vertices have a common color, and no pair of colors is repeated.) - Allan Bickle, Dec 26 2021

Examples

			(a,b)-Fibonacci ratio limits r(a,b) (see a comment above): as a triangle with offset 1 one has t(3, m) = 4 for m = 1, 2, 3. This gives the limits r(a = m,b = A063929(3, m)), i.e., r(1,12) = r(2,8) = r(3,4) = 4 (and the limit 4 appears only for these three (a,b) values). - _Wolfdieter Lang_, Jan 12 2015
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..10000
• Allan Bickle, 2-Tone coloring of joins and products of graphs, Congr. Numer. 217 (2013), 171-190.
• Michael Somos, Sequences used for indexing triangular or square arrays

Crossrefs

Cf. A002024, A002260.

Programs

Magma

[Round(Sqrt(2*(n-1)))+1: n in [2..60]]; // Vincenzo Librandi, Jun 23 2011

Maple

seq(n$(n-1),n=2..15); # Robert Israel, Jan 12 2015

Mathematica

Flatten[Table[PadRight[{},n-1,n],{n,15}]] (* Harvey P. Dale, Feb 26 2012 *)

PARI

t1(n)=floor(3/2+sqrt(2*n-2)) /* A003057 */

PARI

t2(n)=n-1-binomial(floor(1/2+sqrt(2*n-2)),2) /* A002260(n-2) */

Python

from math import isqrt
def A003057(n): return (k:=isqrt(m:=n-1<<1))+int((m<<2)>(k<<2)*(k+1)+1)+1 # Chai Wah Wu, Jul 26 2022

Formula

a(n) = A002260(n) + A004736(n).

a(n) = A002024(n-1) + 1 = floor(sqrt(2*(n - 1)) + 1/2) + 1 = round(sqrt(2*(n - 1))) + 1. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

a(n) = ceiling((sqrt(8*n - 7) + 1)/2). - Reinhard Zumkeller, Aug 28 2001, modified by Frank Ruskey, Nov 06 2007, restored by M. F. Hasler, Jan 13 2015

a(n) = A080036(n-1) - (n - 1) for n >= 2. - Jaroslav Krizek, Jun 19 2009

G.f.: (2*x^2 + Sum_{n>=2} x^(n*(n - 1)/2 + 2))/(1 - x) = (x^2 + x^(15/8)*theta_2(0,sqrt(x))/2)/(1 - x) where theta_2 is the second Jacobi theta function. - Robert Israel, Jan 12 2015

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003