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A037257
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a() = 1,3,... [ A037257 ], differences = 2,... [ A037258 ] and 2nd differences [ A037259 ] are disjoint and monotonic; adjoin next free number to 2nd differences unless it would produce a duplicate in which case ignore.
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15
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1, 3, 9, 20, 38, 64, 100, 148, 209, 284, 374, 480, 603, 745, 908, 1093, 1301, 1533, 1790, 2075, 2389, 2733, 3108, 3515, 3955, 4429, 4938, 5484, 6069, 6694, 7360, 8068, 8819, 9614, 10454, 11340, 12273, 13255, 14287, 15370, 16505, 17693, 18935, 20232
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OFFSET
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0,2
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COMMENTS
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27 and 250 are the first two numbers to be ignored.
I discovered this around 1979; Martin Gardner described a version of it in his 1980 article.
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REFERENCES
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M. Gardner, Weird Numbers from Titan, Isaac Asimov's Science Fiction Magazine, Vol. 4, No. 5, May 1980, pp. 42ff.
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LINKS
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EXAMPLE
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After 1 3 9 20 with differences
------ 2 6 11 and 2nd differences
------- 4 5, the next free number is 7 so we get
----- 1 3 9 20 38 ...
------ 2 6 11 18 ...
------- 4 5 7 ....
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MATHEMATICA
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ClearAll[a]; A037257 = {a[0]=1, a[1]=3, a[2]=9}; d1 = Differences[A037257]; d2 = Differences[d1]; ignored = {}; a[n_] := a[n] = (u = Union[A037257, d1, d2, ignored]; m = MapIndexed[List, u]; sel = Select[m, #1[[1]] != #1[[2, 1]] & , 1]; For[nextFree = sel[[1, 2, 1]], True, nextFree++, an2 = nextFree; an = an2 - a[n-2] + 2*a[n-1]; an1 = an - a[n-1]; If[ FreeQ[ ignored, an2] && Length[ Join[ A037257, d1, d2, {an, an1, an2}]] == Length[ Union[ A037257, d1, d2, {an, an1, an2}]], Break[], AppendTo[ ignored, an2]] ]; AppendTo[ A037257, an]; AppendTo[d1, an1]; AppendTo[d2, an2]; an); Table[a[n], {n, 0, 43}] (* Jean-François Alcover, Sep 14 2012 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000
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STATUS
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approved
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