Archive for integer sequence

shortened iterations [code golf]

Posted in Kids, pictures, Statistics, Travel with tags , , , , , , , , on October 29, 2019 by xi'an

A codegolf lazy morning exercise towards finding the sequence of integers that starts with an arbitrary value n and gets updated by blocks of four as

a_{4k+1} = a_{4k} \cdot(4k+1)\\ a_{4k+2} = a_{4k+1} + (4k+2)\\ a_{4k+3} = a_{4k+2} - (4k+3)\\ a_{4k+4} = a_{4k+3} / (4k+4)

until the last term is not an integer. While the update can be easily implemented with the appropriate stopping rule, a simple congruence analysis shows that, depending on n, the sequence is 4, 8 or 12 values long when

n\not\equiv 1(4)\\ n\equiv 1(4)\ \text{and}\ 3(n-1)+4\not\equiv 0(32)\\ 3(n-1)+4\equiv 0(32)

respectively. But sadly the more interesting fixed length solution

`~`=rep #redefine function
b=(scan()-1)*c(32~4,8,40~4,1,9~3)/32+c(1,1,3,0~3,6,-c(8,1,9,-71,17)/8)
b[!b%%1] #keep integers only

ends up being longer than the more basic one:

a=scan()
while(!a[T]%%1)a=c(a,d<-a[T]*T,d+T+1,e<-d-1,e/((T<-T+4)-1))
a[-T]

where Robin’s suggestion of using T rather than length is very cool as T has double meaning, first TRUE (and 1) then the length of a…