A376004 - OEIS

%I #38 Sep 29 2024 15:11:52

%S 1,1,1,1,1,1,2,1,0,1,1,2,1,0,1,2,1,1,2,0,1,3,2,1,2,-1,0,1,5,3,1,1,-1,

%T -1,-1,1,1,5,3,1,1,-1,-1,-1,1,2,1,4,4,1,2,-2,0,0,1,3,2,1,5,1,2,3,-1,

%U -1,0,1,5,3,1,1,2,2,2,4,-1,-1,-1,1,5,5,3,1,1,3,3,3,-3,-1,-1,-1,1

%N Limiting matrix {m_n}, where m_0 = 1 and m_{i+1} = [[m_i, A(m_i)], [B(m_i), C(m_i)]], read by antidiagonals, and A adds the corresponding x-coords to every element, B subtracts it, and C adds the corresponding y-coords.

%C Start with M_0 = [[1]]. M_n is a 2^n X 2^n matrix. M_{n+1} is constructed from M_n as follows:

%C           | M_n     A(M_n) |

%C M_{n+1} = |                |

%C           | B(M_n)  C(M_n) |

%C A - adds the corresponding x-coords to the elements of the matrix

%C B - subtracts the corresponding x-coords to the elements of the matrix

%C C - adds the corresponding y-coords to the elements of the matrix

%C The coordinate system sets the top-left corner to be (0, 0).

%C Then we take the limiting matrix {M_n}, and turn it into an integer sequence by reading it by antidiagonals.

%H Bryle Morga, <a href="/A376004/b376004.txt">Table of n, a(n) for n = 1..10000</a>

%H Bryle Morga, <a href="/A376004/a376004.png">Visualization of the first 2 million terms.</a>

%e M_0 = [1] by definition. Constructing M_1 goes as follows:

%e A(M_0) = M_0 + [0] = [1]

%e B(M_0) = M_0 - [0] = [1]

%e C(M_0) = M_0 + [0] = [1]

%e So we have:

%e             |  1    1 |

%e M_1 =       |  1    1 |

%e From this M_2 can be constructed:

%e A(M_2) = M_1+[[0, 1],[0, 1]] = [[1, 2], [1, 2]]

%e B(M_2) = M_1-[[0, 1],[0, 1]] = [[1, 0], [1, 0]]

%e C(M_2) = M_1+[[0, 0],[1, 1]] = [[1, 1], [2, 2]]

%e       | 1 1 1 2 |

%e       | 1 1 1 2 |

%e M_2 = | 1 0 1 1 |

%e       | 1 0 2 2 |

%o (Python)

%o def expand(m):

%o   i = len(m)

%o   res = [[0 for _ in range(2*i)] for _ in range(2*i)]

%o   for x in range(i):

%o     for y in range(i):

%o       res[y][x] = m[y][x]

%o       res[y][x+i] = m[y][x] + x

%o       res[y+i][x] = m[y][x] - x

%o       res[y+i][x+i] = m[y][x] + y

%o   return res

%o a = []

%o m = [[1]]

%o for _ in range(11):

%o   m = expand(m)

%o for i in range(len(m)):

%o   for j in range(i+1):

%o     a.append(m[j][i-j])

%K sign,look,tabl

%O 1,7

%A _Bryle Morga_, Sep 05 2024