OFFSET
3,1
COMMENTS
The rules for Pyramid arrangement are: (1) boxes shall be arranged in symmetrical forms; (2) each step width shall be 0.5 or 1, where boxes width = 1.
The number of patterns on each n-th step is A053260(n).
LINKS
Kival Ngaokrajang, Illustration for n = 42, k = 1..8
EXAMPLE
For n = 3..6.
[1] [1] [1|2] [1] [1] [1]
[2|3] [2|3|4] [3|4|5] [2|3|4|5] [2] [2|3]
[3|4|5] [4|5|6]
T(3,1) = 2, T(4,1) = 3, T(5,k) = 0 {no pattern exist due to step width vilolations i.e. [0.5,2], [1.5,1] & [1,0,1]}, T(6,1) = 3, ...
The triangle begins:
n/k 1 2 3 4 5
3 2
4 3
5 0
6 3
7 4
8 4
9 5
10 4
11 5
12 5
13 6 5
14 6
15 6 5
16 7 6
17 6
18 7 6
19 7 6
20 7 7
21 8 7 6
22 8 7 7
23 8 7
24 8 8 7
25 9 8 7
26 8 8 7
27 9 8 8
28 9 8 8 7
29 9 9 8 8
30 9 9 8 8
31 10 9 9 9 8
...
For n = 42, T(n,k) = 11, 11, 11, 10, 11, 10, 10, 9; see illustration in links.
PROG
(Small Basic)
x[0]=1
y[0]=1
for i = 1 To 12
a=math.Power(2, i-1)-2
b=math.Power(2, i)-2
For j = 1 To math.Power(2, i)
m=Math.Remainder(j, 2)
k=math.Round(j/2+(1/2)*m)
y[j+b]=y[k+a]-m+2
x[j+b]=x[k+a]+y[j+b]
EndFor
EndFor
For n = 3 To 50
a[n]=0
c=1
For nn=1 To j+b
If n=x[nn] Then
a[n]=a[n]+1
aa[c]=y[nn]
c=c+1
Else
aa[c]=" "
EndIf
EndFor
For cc=1 To c
TextWindow.Write(aa[cc]+" ")
endfor
TextWindow.WriteLine(" ")
EndFor
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Kival Ngaokrajang, Jul 15 2013
STATUS
approved