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A229037
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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.
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25
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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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.]
OEIS, Pin plot of 200 terms and scatterplot of 10000 terms
Index entries for sequences with interesting graphs or plots
Index entries for non-averaging sequences
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FORMULA
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a(n) <= (n+1)/2. - Charles R Greathouse IV, Jan 21 2014
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MATHEMATICA
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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 *)
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PROG
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(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:
........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
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CROSSREFS
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Cf. A094870.
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.
Sequence in context: A153916 A238597 A045870 * A036863 A209270 A083698
Adjacent sequences: A229034 A229035 A229036 * A229038 A229039 A229040
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KEYWORD
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nonn,easy,nice,look
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AUTHOR
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Jack W Grahl, Sep 11 2013
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STATUS
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approved
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