login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A334932
Numbers that generate rotationally symmetrical XOR-triangles with a pattern of zero-triangles of edge length 3, some of which are clipped to result in some zero-triangles 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
OFFSET
1,1
COMMENTS
Subset of A334769 which is a subset of A334556.
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.
FORMULA
From Colin Barker, Jun 09 2020: (Start)
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(n-2) - 64*a(n-4) 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 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
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, May 16 2020
STATUS
approved