login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.
2

%I #12 Jun 10 2021 15:45:38

%S 1,2,2,2,4,2,6,4,4,6,6,6,4,6,6,6,6,1,1,6,6,15,6,2,9,2,6,15,15,15,2,9,

%T 9,2,15,15,15,15,15,9,9,9,15,15,15,15,15,15,2,9,9,2,15,15,15,15,15,15,

%U 2,4,9,4,2,15,15,15,40,15,15,6,4,4,4,4,6,15,15,40

%N Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.

%C The "Quilt Tiling" is described in Shectman's paper (see Links section).

%C All terms belong to A006498.

%H Rémy Sigrist, <a href="/A345067/b345067.txt">Table of n, a(n) for n = 1..10153</a>

%H J. Parker Shectman, <a href="http://www.ootlinc.com/Fibonacci_Quilt_2_of_3_Cohorts_and_Numeration.pdf">A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration</a>

%H Rémy Sigrist, <a href="/A345067/a345067.png">Illustration of the connection between the "Quilt Tiling" and the sequences A000201 and A005206</a>

%H Rémy Sigrist, <a href="/A345067/a345067.gp.txt">PARI program for A345067</a>

%F T(n, k) = T(k, n).

%F T(n, n) = A130312(n+1)^2.

%F T(n, 1) = A001654(A095791(n)+1).

%F T(n, k) is the square of a Fibonacci number for n = 1+A005206(k+1)..A000201(k).

%e Array T(n, k) begins:

%e n\k| 1 2 3 4 5 6 7 8 9 10 11

%e ---+---+-------+-----------+-------------------+

%e 1 | 1| 2 2| 6 6 6| 15 15 15 15 15|

%e +-----------+ | |

%e 2 | 2| 4 4| 6 6 6| 15 15 15 15 15|

%e | | +---+-------+ |

%e 3 | 2| 4 4| 1| 2 2| 15 15 15 15 15|

%e +---+---+---+---+-------+-------+-----------+

%e 4 | 6 6| 1| 9 9 9| 2 2| 6 6 6|

%e | +---+ +-------+ |

%e 5 | 6 6| 2| 9 9 9| 4 4| 6 6 6|

%e | | | | +---+-------+

%e 6 | 6 6| 2| 9 9 9| 4 4| 1| 2 2|

%e +-------+---+---+-------+-------+---+-------+

%e 7 | 15 15 15| 2| 4 4| 25 25 25 25 25|

%e | | | | |

%e 8 | 15 15 15| 2| 4 4| 25 25 25 25 25|

%e | +---+---+---+ |

%e 9 | 15 15 15| 6 6| 1| 25 25 25 25 25|

%e | | +---+ |

%e 10 | 15 15 15| 6 6| 2| 25 25 25 25 25|

%e | | | | |

%e 11 | 15 15 15| 6 6| 2| 25 25 25 25 25|

%e +-----------+-------+---+-------------------+

%o (PARI) See Links section.

%Y Cf. A000045, A001654, A005206, A006498, A095791, A005206.

%K nonn,look,tabl

%O 1,2

%A _Rémy Sigrist_, Jun 06 2021