A158349 A difference of 2^n Hadamard matrix self-similar types: MatrixCantorSet(2^n) = Hadamard{1,-1}(2^n) - Hadamard{1,0}(2^n).

1, 0, -1, 0, -1, 1, 0, 5, 5, 3, 1, 0, 963, 831, 656, 318, 124, 36, 7, 1, 0, -332914995, -121099959, -54262863, -37433763, -1488468, -4442464, 462362, -241686, 63542, 26, 3732, 776, 113, 45, 1, 1, 0, 155070811326739980630319

Offset

0,8

Comments

Row sums: {1, -1, 0, 14, 2936, -551353600, 497884344990949777256448, ...}.

The Hadamard {1,-1}(2^n) matrices are orthogonal-space-filling.

The Hadamard {1,0}(2^n) matrices are Sierpinski-Pascal/ fractal.

Example matrix:

CantorSet(2^2)={{0, 0, 0, 0},

{0, -1, 0, -1},

{0, -2, -1, 1},

{0, 1, -1, -1}}.

The importance of this set is that the result shows as a polynomial what appears to be a Cantor set of the difference between a space-filling orthogonal set and a fractal set.

As far as I know this is the only reduction of a Cantor set to a polynomial set.

Links

Table of n, a(n) for n=0..38.

Formula

MatrixCantorSet[2^n]=Hadamard{1,-1}(2^n)-Hadamard{1,0}(2^n);

out_(n,m)=coefficients(characteristicpolynomial(MatrixCantorSet[2^n],x),x)

Example

{1},
{0, -1},
{0, -1, 1},
{0, 5, 5, 3, 1},
{0, 963, 831, 656, 318, 124, 36, 7, 1},
{0, -332914995, -121099959, -54262863, -37433763, -1488468, -4442464, 462362, -241686, 63542, 26, 3732, 776, 113, 45, 1, 1},
{0, 155070811326739980630319, 117237547652656472994219, 111108752788151068396112, 59222794789152875507126, 32976901419770952847552, 13735162578113369507264, 5677200005130173303939, 1946514331292045892909, 649464409175484362224, 188852406540173992912, 52960943288584430240, 13296327908790911224, 3203075459142218846, 701613479822512918, 146906825481169552, 28221231795109420, 5162659130637760, 870481803658904, 139113156617630, 20516912834978, 2849268822864, 364528187616, 43483304352, 4755010408, 477088307, 43498607, 3534304, 257390, 15680, 840, 31, 1}

Mathematica

Clear[c, b, An];
Needs["Hadamard`"];
a0 = {1, 2, 4, 8, 16, 32, 64};
a = Join[{{1}}, Table[If[n == 1, Hadamard[2], Hadamard[a0[[n]]][[1]]], {n, 2, 6}]];
c[i_, k_] := Floor[Mod[i/2^k, 2]];
An[d_] := Table[If[Sum[c[n, k]*c[m, k], {k, 0, d - 1}] == 0, 1, 0], {n, 0, d - 1}, {m, 0, d - 1}];
b = Table[An[a0[[n]]], {n, 1, 6}];
c0 = a - b;
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[c0[[ n]], x], x], {n, 1, Length[c]}]];
Flatten[%]
Join[{1}, Table[Apply[Plus, CoefficientList[CharacteristicPolynomial[c[[n]], x], x]], {n, 1, Length[c0]}]];

Crossrefs

Sequence in context: A157703 A332507 A242617 * A320478 A364881 A225302

Adjacent sequences: A158346 A158347 A158348 * A158350 A158351 A158352

Keyword

sign,tabf,uned

Author

Roger L. Bagula, Mar 16 2009

Status

approved