

A003057


n appears n  1 times.


49



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
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OFFSET

2,1


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 n1 <= C(a(n),2).  Frank Ruskey, Nov 06 2007
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, ...
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


LINKS



FORMULA

a(n) = A002024(n1) + 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
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


EXAMPLE

(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


MAPLE



MATHEMATICA

Flatten[Table[PadRight[{}, n1, n], {n, 15}]] (* Harvey P. Dale, Feb 26 2012 *)


PROG

(PARI) t1(n)=floor(3/2+sqrt(2*n2)) /* A003057 */
(PARI) t2(n)=n1binomial(floor(1/2+sqrt(2*n2)), 2) /* A002260(n2) */
(Python)
from math import isqrt
def A003057(n): return (k:=isqrt(m:=n1<<1))+int((m<<2)>(k<<2)*(k+1)+1)+1 # Chai Wah Wu, Jul 26 2022


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

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


STATUS

approved



