A355680 Numerator generator for offsets from the quarter points of the Cantor ternary set to the center points of deleted middle thirds: 1 is in the list and if m is in the list -3m-4 and -3m+4 are in the list, which is ordered by absolute value.

1, -7, 17, 25, -47, -55, -71, -79, 137, 145, 161, 169, 209, 217, 233, 241, -407, -415, -431, -439, -479, -487, -503, -511, -623, -631, -647, -655, -695, -703, -719, -727, 1217, 1225, 1241, 1249, 1289, 1297, 1313, 1321, 1433, 1441, 1457, 1465, 1505, 1513, 1529, 1537, 1865

Offset

1,2

Comments

At the (k+1)-th step of generating the Cantor set, the offsets from 1/4 to the center points of the deleted middle thirds are {a(i)/(4*(-3)^k) : 1 <= i <= 2^k}. Clearly, these offsets are negated for use with respect to 3/4.

Note that each quarter point of the Cantor ternary set, C, is also a quarter point of an interval-constrained subset of C that is an image of C scaled by 3^(-k) for all k >= 1.

If we replace -3m-4 and -3m+4 in the definition with -3m-2 and -3m+2 we get the terms of A191108 and their negation.

Links

Table of n, a(n) for n=1..49.

Eric Weisstein's World of Mathematics, Cantor Set.

Example

At the 2nd step of generating the Cantor set, the deleted middle thirds are (1/9, 2/9) and (7/9, 8/9) with center points 1/6 and 5/6. These points are offset from 1/4 by -1/12 and +7/12. The denominator for the 2nd step (i.e., k=1) is 4*(-3)^k = -12. So a(1) = -1 * -1 = 1 and a(2) = 7 * -1 = -7.

Prog

(PARI) A355680(size) = {a=vector(size); a[1] = 1;
forstep (n=2, size, 2, j=-3*a[n\2];
  if(j>0, a[n-1]=j-4; a[n]=j+4, a[n-1]=j+4; a[n]=j-4);
  print(n-1, " ", a[n-1]); print(n, " ", a[n]); ) }

Crossrefs

Essentially, the positions of isolated 0's in A355682.

Cf. A191108.

Sequence in context: A309231 A183897 A300186 * A354672 A261934 A253075

Adjacent sequences: A355677 A355678 A355679 * A355681 A355682 A355683

Keyword

sign,easy

Author

Peter Munn, Jul 14 2022

Status

approved