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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.

Examples

			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 ....
		

References

• M. Gardner, Weird Numbers from Titan, Isaac Asimov's Science Fiction Magazine, Vol. 4, No. 5, May 1980, pp. 42ff.

Crossrefs

Cf. A005228, A030124.

Programs

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 *)

Extensions

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