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 A135642 Concave numbers. 12
 110, 120, 121, 122, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 145, 146, 150, 151, 152, 153, 154, 155, 156, 157, 158, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The structure of digits represent a concave function or a concave object. In the graphic representation the points are connected by imaginary line segments (or line curves) from left to right. Only strictly concave numbers are included in this sequence; the interpolation between at least one pair of digits must be strictly less than some intermediate digit. - Franklin T. Adams-Watters, Jan 26 2014 LINKS Rémy Sigrist, Table of n, a(n) for n = 1..10000 EXAMPLE The number 12221 is a concave number. Note that the number of this sequence (A135642) is also a concave number as shown below: . 9     . . . . . .      . . . . . . 8     . . . . . .      . . . . . . 7     . . . . . .      . . . . . . 6     . . . x . .      . . . 6 . . 5     . . x . . .      . . 5 . . . 4     . . . . x .      . . . . 4 . 3     . x . . . .      . 3 . . . . 2     . . . . . x      . . . . . 2 1     x . . . . .      1 . . . . . 0     . . . . . .      . . . . . . . Another example is 1342. On the other hand 3124 is not in the sequence, it's in A135641. 1234 is not in the sequence, it's in A135643. 1243 is not in the sequence, it's in A163278. - Omar E. Pol, Jan 29 2014 MATHEMATICA concaveQ[n_] := With[{dd = IntegerDigits[n]}, AllTrue[SequencePosition[dd, {_, _, _}][[All, 1]], dd[[#]] + dd[[#+2]] < 2 dd[[#+1]]&]]; Select[Range[100, 200], concaveQ] (* Jean-François Alcover, Nov 01 2018 *) PROG (PARI) { isconcave(n) = my(t, r); t=eval(Vec(Str(n))); r=0; for(i=1, #t, for(j=i+2, #t, for(k=i+1, j-1, if( t[k]*(j-i) < t[i]*(j-k) + t[j]*(k-i), return(0)); if( t[k]*(j-i) > t[i]*(j-k) + t[j]*(k-i), r=1); ))); r } /* Franklin T. Adams-Watters and Max Alekseyev, Jan 30 2014 */ CROSSREFS Cf. A135600, A135601, A135602, A135603, A135641, A135643, A163278. Sequence in context: A036230 A171236 A279421 * A166323 A194429 A320121 Adjacent sequences:  A135639 A135640 A135641 * A135643 A135644 A135645 KEYWORD nonn,base AUTHOR Omar E. Pol, Nov 30 2007 STATUS approved

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Last modified March 23 14:17 EDT 2019. Contains 321431 sequences. (Running on oeis4.)