

A229037


The "forest fire": sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression.


32



1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 9, 4, 4, 5, 5, 10, 5, 5, 10, 2, 10, 13, 11, 10, 8, 11, 13, 10, 12, 10, 10, 12, 10, 11, 14, 20, 13
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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
a(A241752(n)) = n and a(m) != n for m < A241752(n).  Reinhard Zumkeller, Apr 28 2014


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)
Jean Bickerton, Colored plot of 200000 terms; color is difference from previous term in sequence
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.]
OEIS, Pin plot of 200 terms and scatterplot of 10000 terms
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 20190821), 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 and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Index entries for sequences with interesting graphs or plots
Index entries for nonaveraging sequences


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[nk]  a[n2*k], Throw@True], {k, Floor[(n1)/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[nd]v[n2*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 !! (n1)
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 n2*i >= 0:
........b.add(2*A229037_list[ni]A229037_list[n2*i])
........i += 1
........while j in b:
............b.remove(j)
............j += 1
....A229037_list.append(j) # Chai Wah Wu, Dec 21 2014


CROSSREFS

Cf. A094870.
For a variant see A309890.
A selection of sequences related to "no threeterm arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.
Sequence in context: A238597 A045870 A309890 * A036863 A209270 A083698
Adjacent sequences: A229034 A229035 A229036 * A229038 A229039 A229040


KEYWORD

nonn,easy,nice,look


AUTHOR

Jack W Grahl, Sep 11 2013


STATUS

approved



