OFFSET
1,3
COMMENTS
Added name "forest fire" to make it easier to locate this sequence. - N. J. A. Sloane, Sep 03 2019
This sequence and A235383 and A235265 were winners in the best new sequence contest held at the OEIS Foundation booth at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
See A236246 for indices n such that a(n)=1. - M. F. Hasler, Jan 20 2014
See A241673 for indices n such that a(n)=2^k. - Reinhard Zumkeller, Apr 26 2014
The graph (for up to n = 10000) has an eerie similarity (why?) to the distribution of rising smoke particles subjected to a lateral wind, and where the particles emanate from randomly distributed burning areas in a fire in a forest or field. - Daniel Forgues, Jan 21 2014
The graph (up to n = 100000) appears to have a fractal structure. The dense areas are not random but seem to repeat, approximately doubling in width and height each time. - Daniel Forgues, Jan 21 2014
The indices n such that a(n) = 1 are given by A236313 (relative spacing) up to 19 terms, and A003278 (directly) up to 20 terms. If just placing ones, the 21st 1 would be n=91. The sequence A003278 fails at n=91 because the numbers filling the gaps create an arithmetic progression with a(27)=9, a(59)=5 and a(91)=1. Additionally, if you look at indices n starting at the first instance of 4 or 5, A003278/A236313 provide possible indices for a(n)=4 or a(n)=5, with some indexes instead filled by numbers < (4,5). A003278/A236313 also fail to predict indices for a(n)=4 or a(n)=5 by the ~20th term. - Daniel Putt, Sep 29 2022
LINKS
Giovanni Resta, Alois P. Heinz, and Charles R Greathouse IV, Table of n, a(n) for n = 1..100000 (1..1000 from Resta, 1001..10000 from Heinz, and 10001..100000 from Greathouse)
Xan Gregg, Enhanced scatterplot of 10000 terms [In this plot, the points have been made translucent to reduce the information lost to overstriking, and the point size varies with n in an attempt to keep the density comparable.]
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), MathOverflow
Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
Sébastien Palcoux, Density plot of the first 1000000 terms
Reddit User "garnet420", Colored plot of 16 million terms; horizontal divisions are 1000000; vertical divisions are 25000
Reddit User "garnet420", B/W plot of 16 million terms; horizontal divisions are 1000000; vertical divisions are 25000
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
FORMULA
a(n) <= (n+1)/2. - Charles R Greathouse IV, Jan 21 2014
MATHEMATICA
a[1] = 1; a[n_] := a[n] = Block[{z = 1}, While[Catch[ Do[If[z == 2*a[n-k] - a[n-2*k], Throw@True], {k, Floor[(n-1)/2]}]; False], z++]; z]; a /@ Range[100] (* Giovanni Resta, Jan 01 2014 *)
PROG
(PARI) step(v)=my(bad=List(), n=#v+1, t); for(d=1, #v\2, t=2*v[n-d]-v[n-2*d]; if(t>0, listput(bad, t))); bad=Set(bad); for(i=1, #bad, if(bad[i]!=i, return(i))); #bad+1
first(n)=my(v=List([1])); while(n--, listput(v, step(v))); Vec(v) \\ Charles R Greathouse IV, Jan 21 2014
(Haskell)
import Data.IntMap (empty, (!), insert)
a229037 n = a229037_list !! (n-1)
a229037_list = f 0 empty where
f i m = y : f (i + 1) (insert (i + 1) y m) where
y = head [z | z <- [1..],
all (\k -> z + m ! (i - k) /= 2 * m ! (i - k `div` 2))
[1, 3 .. i - 1]]
-- Reinhard Zumkeller, Apr 26 2014
(Python)
A229037_list = []
for n in range(10**6):
i, j, b = 1, 1, set()
while n-2*i >= 0:
i += 1
while j in b:
b.remove(j)
j += 1
A229037_list.append(j) # Chai Wah Wu, Dec 21 2014
CROSSREFS
For a variant see A309890.
KEYWORD
AUTHOR
Jack W Grahl, Sep 11 2013
STATUS
approved