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.

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

Offset: 1

Author

Jack W Grahl, Sep 11 2013

Keywords

• nonn
• easy
• nice
• look

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

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)
• 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: 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)
• Index entries for sequences with interesting graphs or plots
• Index entries for non-averaging sequences

Crossrefs

Cf. A094870, A362942 (partial sums).

For a variant see A309890.

A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

Programs

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

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 *)

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

Python

A229037_list = []
for n in range(10**6):
    i, j, b = 1, 1, set()
    while n-2*i >= 0:
        b.add(2*A229037_list[n-i]-A229037_list[n-2*i])
        i += 1
        while j in b:
            b.remove(j)
            j += 1
    A229037_list.append(j) # Chai Wah Wu, Dec 21 2014

Formula

a(n) <= (n+1)/2. - Charles R Greathouse IV, Jan 21 2014