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A191108
Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 3x+2 are in a.
9
1, 5, 13, 17, 37, 41, 49, 53, 109, 113, 121, 125, 145, 149, 157, 161, 325, 329, 337, 341, 361, 365, 373, 377, 433, 437, 445, 449, 469, 473, 481, 485, 973, 977, 985, 989, 1009, 1013, 1021, 1025, 1081, 1085, 1093, 1097, 1117, 1121, 1129, 1133, 1297, 1301, 1309, 1313, 1333, 1337, 1345, 1349, 1405, 1409, 1417, 1421, 1441, 1445
OFFSET
1,2
COMMENTS
See discussions at A190803, A191106. The sequence a=A191108 has closure properties: the positive integers in (2+A191108)/3 comprise A191108, as do those in (-2+A191108)/3.
From Peter Munn, May 13 2019: (Start)
The closure of {1} in the positive integers under reflection about 3^k, k >= 1.
Asymptotic density is 0.
Consider a Sierpinski arrowhead curve formed of edges numbered consecutively from 0 at its axis of symmetry. The m-th edge is contained in the boundary of the plane sector occupied by the arrowhead if and only if m or -m is in this sequence.
For k >= 0, a(2^k) = 2*3^k - 1 and {a(i)/(2*3^k) | 1 <= i <= 2^k} is the set of center points of surviving intervals at the k-th step of generating the Cantor set, and therefore the set of center points of deleted middle-third intervals at the (k+1)-th step.
Define t: Z -> P(R) so that t(n) is the translated Cantor ternary set spanning [(n-1)/2, (n+1)/2], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3.
(End)
LINKS
Eric Weisstein's World of Mathematics, Cantor Set
FORMULA
From Peter Munn, May 25 2019: (Start)
a(n) = (A055247(2n-1) + A055247(2n)) / 3.
a(n) = A306556(2n)*2 - 1 = A306556(2n-1) + A306556(2n).
a(n) = 2*A005823(n) + 1 = 4*A005836(n) + 1 = 2*A191106(n) - 1.
a(2^k+i) = 2*A147991(2^k+i-1) + 3^(k+1) for k >= 0, 1 <= i <= 2^k.
(End)
MATHEMATICA
h = 3; i = -2; j = 3; k = 2; f = 1; g = 7;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]] (* A191108 *)
b = (a + 2)/3; c = (a - 2)/3; r = Range[1, 900];
d = Intersection[b, r] (* A191108 closure property *)
e = Intersection[c, r] (* A191108 closure property *)
PROG
(PARI) a(n) = fromdigits(binary(n-1), 3)<<2 + 1; \\ Kevin Ryde, Aug 05 2022
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 26 2011
STATUS
approved