OFFSET

1,1

COMMENTS

Numbers m in this sequence A070939(m) (mod 3) = 2. The numbers in this sequence can be constructed using run lengths of bits.

2n has the reverse run length pattern as 2n - 1. a(1) has the run lengths {1, 2, 1, 1, 3}, while a(2) has {3, 1, 1, 2, 1}, etc.

For n = 1 (mod 8): 12..(1132)..113;

For n = 3 (mod 8): 113..(2113)..2112;

For n = 5 (mod 8): 11123..(1123)..1122;

For n = 7 (mod 8): 123112..(3112)..31123, where the parenthetic run lengths occur, when they occur, in multiples of 3. Thus, a(9) has the run length form 12113211321132113 = binary 10010111001011100101110010111 = decimal 317049751.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..1264

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

Michael De Vlieger, Central zero-triangles in rotationally symmetrical XOR-Triangles, 2020.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2097153,0,0,0,0,0,0,0,-2097152).

FORMULA

From Alejandro J. Becerra Jr., Mar 01 2021: (Start)

G.f.: x*(1654784*x^13 + 1359872*x^12 + 477184*x^11 + 1236992*x^10 + 1733632*x^9 + 379648*x^8 + 941241*x^7 + 644007*x^6 + 13029*x^5 + 10707*x^4 + 1693*x^3 + 1483*x^2 + 233*x + 151)/((1 - x^8)*(1 - 2097152*x^8)).

a(n) = 2097153*a(n-8) - 2097152*a(n-16). (End)

EXAMPLE

Diagrams of a(1)-a(6), replacing "0" with "." and "1" with "@" for clarity:

a(1) = 151 (a(2) = 233 appears as a mirror image):

@ . . @ . @ @ @

@ . @ @ @ . .

@ @ . . @ .

. @ . @ @

@ @ @ .

. . @

. @

@

.

a(3) = 1483 (a(4) = 1693 appears as a mirror image):

@ . @ @ @ . . @ . @ @

@ @ . . @ . @ @ @ .

. @ . @ @ @ . . @

@ @ @ . . @ . @

. . @ . @ @ @

. @ @ @ . .

@ . . @ .

@ . @ @

@ @ .

. @

@

.

a(5) = 10707 (a(6) = 13029 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] & @@ {#1, Which[#2 == 1, PadRight[{1, 2}, 12 Ceiling[#1/8] - 7, {3, 2, 1, 1}], #2 == 2, PadRight[{1, 1}, 12 Ceiling[#1/8] - 6, {1, 1, 3, 2}]~Join~{2}, #2 == 3, PadRight[{1, 1}, 12 Ceiling[#1/8] - 4, {3, 1, 1, 2}]~Join~{2}, True, PadRight[{}, 12 Ceiling[#1/8] - 1, {1, 2, 3, 1}]]} & @@ {#, Ceiling[Mod[#, 8]/2]} &, 22]

(* 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 &]]

(* From G.f.: *)

Rest@ CoefficientList[Series[x (1654784 x^13 + 1359872 x^12 + 477184 x^11 + 1236992 x^10 + 1733632 x^9 + 379648 x^8 + 941241 x^7 + 644007 x^6 + 13029 x^5 + 10707 x^4 + 1693 x^3 + 1483 x^2 + 233 x + 151)/((1 - x^8) (1 - 2097152 x^8)), {x, 0, 22}], x] (* Michael De Vlieger, Mar 19 2021 *)

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Michael De Vlieger, May 16 2020

STATUS

approved