Pensieve header: Counting classical braids by computing $\bar\Gamma$ of all classical proud braids.

SetDirectory@"C:\\drorbn\\AcademicPensieve\\Projects\\OU";
<< "OU-Programs.m"

2

Timing@CountBs[2,0]

Timing@CountBs[2,1]

Timing@CountBs[2,2]

Timing@CountBs[2,3]

3

Timing@CountBs[3,0]

Timing@CountBs[3,1]

Timing@CountBs[3,2]

Timing@CountBs[3,3]

Timing@CountBs[3,4]

Timing@CountBs[3,5]

Timing@CountBs[3,6]

Timing@CountBs[3,7]

Timing@CountBs[3,8]

Timing@CountBs[3,9]

Timing@CountBs[3,10]

Timing@CountBs[3,11]

Timing@CountBs[3,12]

f3[m]=FullSimplify[m//FindSequenceFunction@{1,5,17,47,115,263,577,1233,2589,5371}]

Expand[f3[m]/.m->11]

Expand[f3[m]/.m->12]

Table[12×(2^m)-2Fibonacci[m+5]-2m-1, {m, 0,12}]

Table[12×(2^m)-2Fibonacci[m+5]-2m-1, {m, 0,15}]//FindLinearRecurrence

FindGeneratingFunction[Table[12×(2^m)-2Fibonacci[m+5]-2m-1, {m, 0,15}],x]//Factor

4

Timing@CountBs[4,0]

Timing@CountBs[4,1]

Timing@CountBs[4,2]

Timing@CountBs[4,3]

Timing@CountBs[4,4]

Timing@CountBs[4,5]

Timing@CountBs[4,6]

Timing@CountBs[4,7]

Timing@CountBs[4,8]

Timing@CountBs[4,9]

5

Timing@CountBs[5,0]

Timing@CountBs[5,1]

Timing@CountBs[5,2]

Timing@CountBs[5,3]

Timing@CountBs[5,4]

Timing@CountBs[5,5]

Timing@CountBs[5,6]

Timing@CountBs[5,7]

Timing@CountBs[5,8]

Timing@CountBs[5,9]

6

Timing@CountBs[6,0]

Timing@CountBs[6,1]

Timing@CountBs[6,2]

Timing@CountBs[6,3]

Timing@CountBs[6,4]

Timing@CountBs[6,5]

Timing@CountBs[6,6]

Timing@CountBs[6,7]

Timing@CountBs[6,8]

Timing@CountBs[6,9]

7

Timing@CountBs[7,0]

Timing@CountBs[7,1]

Timing@CountBs[7,2]

Timing@CountBs[7,3]

Timing@CountBs[7,4]

Timing@CountBs[7,5]

Timing@CountBs[7,6]

Timing@CountBs[7,7]

Timing@CountBs[7,8]

8

Timing@CountBs[8,0]

Timing@CountBs[8,1]

Timing@CountBs[8,2]

Timing@CountBs[8,3]

Timing@CountBs[8,4]

Timing@CountBs[8,5]

Timing@CountBs[8,6]

Timing@CountBs[8,7]

9

Timing@CountBs[9,0]

Timing@CountBs[9,1]

Timing@CountBs[9,2]

Timing@CountBs[9,3]

Timing@CountBs[9,4]

Timing@CountBs[9,5]

Timing@CountBs[9,6]

Timing@CountBs[9,7]

n

Expand[n-1//FindSequenceFunction@{5,17,33,53,77}]

Expand[n-2//FindSequenceFunction@{47,131,259,439,679,987,1371}]

-21-((22 n)/3)+6 (n^2)+((4 (n^3))/3)/.n->3