A037257 - OEIS

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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.

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

Table of n, a(n) for n=0..43.

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: A348090 A139142 A225385 * A145068 A293357 A202349

Adjacent sequences: A037254 A037255 A037256 * A037258 A037259 A037260

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

STATUS

approved