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 A037257 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. 15
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. REFERENCES M. Gardner, Weird Numbers from Titan, Isaac Asimov's Science Fiction Magazine, Vol. 4, No. 5, May 1980, pp. 42ff. LINKS EXAMPLE 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 .... MATHEMATICA 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 *) CROSSREFS Cf. A005228, A030124. Sequence in context: A037048 A139142 A225385 * A145068 A293357 A202349 Adjacent sequences:  A037254 A037255 A037256 * A037258 A037259 A037260 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000 STATUS approved

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Last modified January 27 05:41 EST 2021. Contains 340455 sequences. (Running on oeis4.)