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A334930 Numbers that generate rotationally symmetrical XOR-triangles featuring singleton zero bits in a hexagonal arrangement. 3
1, 11, 13, 91, 109, 731, 877, 5851, 7021, 46811, 56173, 374491, 449389, 2995931, 3595117, 23967451, 28760941, 191739611, 230087533, 1533916891, 1840700269, 12271335131, 14725602157, 98170681051, 117804817261, 785365448411, 942438538093, 6282923587291, 7539508304749 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Subset of A334556.
No zero appears in the center of the figure, thus a(n) does not intersect A334769.
Numbers m with A070939(m) (mod 3) = 1 involving alternating run lengths of a singleton zero separated by a pair of 1s in the binary expansion, admitting an initial or final singleton 1.
LINKS
Michael De Vlieger, Diagram montage of XOR-triangles resulting from a(n) with 2 <= n <= 33.
FORMULA
G.f.: (1 + 11*x + 4*x^2 - 8*x^3)/(1 - 9*x^2 + 8*x^4).
a(n) = - (4/7) - (1/7)*(-1)^(n-1) + ((6 + 10*sqrt(2))/7)*(2*sqrt(2))^(n-1) + ((6 - 10*sqrt(2))/7)*(-2*sqrt(2))^(n-1) - Alejandro J. Becerra Jr., May 31 2020
EXAMPLE
Diagrams of a(2)-a(5), replacing “0” with “.” and “1” with “@” for clarity:
a(2)=11 a(3)=13
@ . @ @ @ @ . @
@ @ . . @ @
. @ @ .
@ @
.
a(4) = 91 a(5) = 109
@ . @ @ . @ @ @ @ . @ @ . @
@ @ . @ @ . . @ @ . @ @
. @ @ . @ @ . @ @ .
@ . @ @ @ @ . @
@ @ . . @ @
. @ @ .
@ @
MATHEMATICA
CoefficientList[Series[(1 + 11 x + 4 x^2 - 8 x^3)/(1 - 9 x^2 + 8 x^4), {x, 0, 28}], x]
(* Generate a textual plot of XOR-triangle T(n) *)
xortri[n_Integer] := TableForm@ MapIndexed[StringJoin[ConstantArray[" ", First@ #2 - 1], StringJoin @@ Riffle[Map[If[# == 0, "." (*0*), "@" (*1*)] &, #1], " "]] &, NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[n, 2], Length@ # > 1 &]]
CROSSREFS
Sequence in context: A056446 A023317 A056436 * A089824 A276694 A086549
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, May 16 2020
STATUS
approved

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Last modified February 20 20:16 EST 2024. Contains 370217 sequences. (Running on oeis4.)