#helppls
modest crowBOT
dusky carbon
@narrow depot
narrow depot
show more context, like the material where they defined tilings
dusky carbon
@narrow depot it is a dominoes problem, where normally you can tile 1 by j with dominoes and monominoes, but know we are increasing the size with odd values.
narrow depot
I need a definition of tiling to help
dusky carbon
tiling is the act of covering 1 by n board with 1 by k tiles such that there i no cross over or intersection,
In our case we are restricted with 1 by n and 1 by k
narrow depot
j and n are whole numbers?
covering is complete?
no rotations allowed?
if yes to all 3, this looks just like partitions of integers.
but I assume order matters?
e.g. tiling a 1x5 with 1x1s, 1x3s, and 1x5s could be:
[5]
[3,1,1,], [1,3,1], [1,1,3]
[1,1,1,1,1]
narrow depot
well there is an obvious recurrence relation
if f(j) is odd tillings of a 1xj,
f(j) = f(j-1) + f(j-3) + ...
it can be an infinite f(a) for a <0 is 0
but you could make it finite by going to 2floor(0.5+j/2)
f(8) = f(7) + f(5) + f(3) + f(1) = 21
f(7) = f(6) + f(4) + f(2) + f(0) = 13
f(6) = f(5) + f(3) + f(1) = 8
f(5) = f(4) + f(2) + f(0) = 5
f(4) = f(3) + f(1) = 3
f(3) = f(2) + f(0) = 2
f(2) = f(1) = 1
f(1) = 1
f(0) = 1
easy
lets look at f(8) and f(6),
f(8) = f(7) + f(5) + f(3) + f(1)
f(6) = f(5) + f(3) + f(1)
f(8) = f(7) + f(6)
I'd split j for odd and even and prove inductively
you gotta show a little more rigor but yes
dusky carbon
is not that insane tho ?
narrow depot
no, golden spirals are everywhere
dusky carbon
But how geometrically a golden spiral would work in a 1 by j board @narrow depot
liek in smth liek that:
narrow depot
it's everywhere, the dimension doesn't matter