login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A096810 Fractal table, read by antidiagonals, consisting of numbers 0..3. 2

%I #8 Jun 13 2017 22:10:09

%S 0,2,0,2,1,0,2,1,2,0,2,2,1,1,0,2,1,3,0,2,0,2,2,1,1,2,1,0,2,1,2,1,2,1,

%T 2,0,2,2,2,1,1,2,1,1,0,2,1,3,0,3,0,3,0,2,0,2,2,1,2,3,1,0,1,2,1,0,2,1,

%U 2,1,3,1,2,0,2,1,2,0,2,2,2,1,1,2,1,1,2,2,1,1,0,2,1,3,0,2,1,3,0,2,1,3,0,2,0

%N Fractal table, read by antidiagonals, consisting of numbers 0..3.

%C Antidiagonal sums form A007494 (numbers congruent to 0 or 2 mod 3). Sums of squares of antidiagonals form A096808 and is congruent to A004526 mod 4. Using these terms as powers of (-1) results in the table of signs A096809.

%F For n>=0: T(0, n)=0, T(n+1, 0)=2, T(n+1, n+1)=1. T(k, n) = 3 - T(n, k) for n>0, k>=0 and n != k. Construction: start with T(0, 0)=0 and proceed for all i>=0 in this way: for k=0..2^i-1, concatenate the (2^i)x(2^i) matrix to itself to form a matrix twice its size: T(n, k+2^i)=T(n, k), T(n+2^i, k)=T(n, k), T(n+2^i, k+2^i)=T(n, k); then for n=0..2^i-1, increment these elements by +1: T(2^i, n), T(n+2^i, n), T(n+2^i, 2^i). Example: start with the matrix: 0 0 2 1 concatenate this matrix to itself to form a matrix twice the size: 0 0 | 0 0 2 1 | 2 1 ----+---- 0 0 | 0 0 2 1 | 2 1 then increment the elements that comprise the far left column of the matrix in the lower right quadrant and those elements that comprise the top row and diagonal of the matrix in the lower left quadrant (the element found in both the top row and diagonal gets incremented twice): 0 0 | 0 0 2 1 | 2 1 ----+---- 2 1 | 1 0 2 2 | 3 1 Repeating these steps forms this table.

%e The elements in the table begin:

%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

%e 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1

%e 2 1 1 0 2 1 1 0 2 1 1 0 2 1 1 0

%e 2 2 3 1 2 2 3 1 2 2 3 1 2 2 3 1

%e 2 1 1 1 1 0 0 0 2 1 1 1 1 0 0 0

%e 2 2 2 1 3 1 2 1 2 2 2 1 3 1 2 1

%e 2 1 2 0 3 1 1 0 2 1 2 0 3 1 1 0

%e 2 2 3 2 3 2 3 1 2 2 3 2 3 2 3 1

%e 2 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

%e 2 2 2 1 2 1 2 1 3 1 2 1 2 1 2 1

%e 2 1 2 0 2 1 1 0 3 1 1 0 2 1 1 0

%e 2 2 3 2 2 2 3 1 3 2 3 1 2 2 3 1

%e 2 1 1 1 2 0 0 0 3 1 1 1 1 0 0 0

%e 2 2 2 1 3 2 2 1 3 2 2 1 3 1 2 1

%e 2 1 2 0 3 1 2 0 3 1 2 0 3 1 1 0

%e 2 2 3 2 3 2 3 2 3 2 3 2 3 2 3 1

%e The sum of the antidiagonals begin: {0,2,3,5,6,8,9,11,12,14,...}.

%o (PARI) T(n,k)=local(M,D=6);if(n<0 || k<0,0, M=matrix(2^D,2^D);M[2,1]=2;M[2,2]=1; for(i=1,D-1, for(r=1,2^i, for(c=1,2^i, M[r,c+2^i]=M[r,c];M[r+2^i,c]=M[r,c];M[r+2^i,c+2^i]=M[r,c]); M[1+2^i,r]+=1;M[r+2^i,r]+=1;M[r+2^i,1+2^i]+=1;));M[n+1,k+1])

%Y Cf. A096808, A096809, A007494, A003987.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Jul 21 2004

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)