|
%I #75 Jul 27 2022 02:20:09
%S 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,
%T 9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12,
%U 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
%N n appears n - 1 times.
%C 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
%C Smallest integer such that n-1 <= C(a(n),2). - _Frank Ruskey_, Nov 06 2007
%C a(n) = inverse (frequency distribution) sequence of A161680. - _Jaroslav Krizek_, Jun 19 2009
%C 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
%C Square array, T(n,k) = n + k + 2, n > = 0 and k >= 0, read by antidiagonals. Northwest corner:
%C 2, 3, 4, 5, ...
%C 3, 4, 5, 6, ...
%C 4, 5, 6, 7, ...
%C 5, 6, 7, 8, ...
%C ... - _Franck Maminirina Ramaharo_, Nov 21 2018
%C 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
%H Vincenzo Librandi, <a href="/A003057/b003057.txt">Table of n, a(n) for n = 2..10000</a>
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/2tonejcpaper.pdf">2-Tone coloring of joins and products of graphs</a>, Congr. Numer. 217 (2013), 171-190.
%H Michael Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>
%F a(n) = A002260(n) + A004736(n).
%F 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
%F 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
%F a(n) = A080036(n-1) - (n - 1) for n >= 2. - _Jaroslav Krizek_, Jun 19 2009
%F 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
%e (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
%p seq(n$(n-1),n=2..15); # _Robert Israel_, Jan 12 2015
%t Flatten[Table[PadRight[{},n-1,n],{n,15}]] (* _Harvey P. Dale_, Feb 26 2012 *)
%o (PARI) t1(n)=floor(3/2+sqrt(2*n-2)) /* A003057 */
%o (PARI) t2(n)=n-1-binomial(floor(1/2+sqrt(2*n-2)),2) /* A002260(n-2) */
%o (Magma) [Round(Sqrt(2*(n-1)))+1: n in [2..60]]; // _Vincenzo Librandi_, Jun 23 2011
%o (Python)
%o from math import isqrt
%o 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
%Y Cf. A002024, A002260.
%K nonn,easy
%O 2,1
%A _N. J. A. Sloane_
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003
|
