

A334932


Numbers that generate rotationally symmetrical XORtriangles with a pattern of zerotriangles of edge length 3, some of which are clipped to result in some zerotriangles of edge length 2 at the edges.


3



2535, 3705, 162279, 237177, 10385895, 15179385, 664697319, 971480697, 42540628455, 62174764665, 2722600221159, 3979184938617, 174246414154215, 254667836071545, 11151770505869799, 16298741508578937, 713713312375667175, 1043119456549052025, 45677651992042699239
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Numbers m in this sequence A070939(m) (mod 3) = 0. All m have first and last bits = 1.
The numbers in this sequence can be constructed using run lengths of bits thus: 12..(42)..3 or the reverse 3..(24)..21, with at least one copy of the pair of parenthetic numbers.
Thus, the smallest number m has run lengths {1, 2, 4, 2, 3}, which is the binary 100111100111 = decimal 2535.
2n has the reverse run length pattern as 2n  1. a(3) has the run lengths {1, 2, 4, 2, 4, 2, 3}, while a(4) has {3, 2, 4, 2, 4, 2, 1}, etc.


LINKS

Michael De Vlieger, Diagram montage of XORtriangles resulting from a(n) with 1 <= n <= 32.


FORMULA

G.f.: 3*x*(13 + 19*x)*(65  64*x^2) / ((1  x)*(1 + x)*(1  8*x)*(1 + 8*x)).
a(n) = 65*a(n2)  64*a(n4) for n>4.
a(n) = (1/21)*(16  3*(1)^n + 123*2^(5+3*n)  85*(1)^n*2^(5 + 3*n)) for n>0.
(End)


EXAMPLE

Diagrams of a(1)a(4), replacing “0” with “.” and “1” with “@” for clarity:
a(1) = 2535 (a(2) = 3705 appears as a mirror image):
@ . . @ @ @ @ . . @ @ @
@ . @ . . . @ . @ . .
@ @ @ . . @ @ @ @ .
. . @ . @ . . . @
. @ @ @ @ . . @
@ . . . @ . @
@ . . @ @ @
@ . @ . .
@ @ @ .
. . @
. @
@
.
a(3) = 162279 (a(4) = 237177 appears as a mirror image):
@ . . @ @ @ @ . . @ @ @ @ . . @ @ @
@ . @ . . . @ . @ . . . @ . @ . .
@ @ @ . . @ @ @ @ . . @ @ @ @ .
. . @ . @ . . . @ . @ . . . @
. @ @ @ @ . . @ @ @ @ . . @
@ . . . @ . @ . . . @ . @
@ . . @ @ @ @ . . @ @ @
@ . @ . . . @ . @ . .
@ @ @ . . @ @ @ @ .
. . @ . @ . . . @
. @ @ @ @ . . @
@ . . . @ . @
@ . . @ @ @
@ . @ . .
@ @ @ .
. . @
. @
@


MATHEMATICA

Array[FromDigits[Flatten@ MapIndexed[ConstantArray[#2, #1] & @@ {#1, Mod[First[#2], 2]} &, If[EvenQ@ #1, Reverse@ #2, #2]], 2] & @@ {#, Join[{1, 2}, PadRight[{}, Ceiling[#, 2], {4, 2}], {3}]} &, 19]
(* Generate a textual plot of XORtriangle 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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



